Pythagorean Identities - Trigonometry

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Question

$\frac{1}{(cos (33^{\circ}))^{2}} - (tan (33^{\circ}))^{2} = ?$

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Answer

$\frac{1}{(cos (33^{\circ}))^{2}} - (tan (33^{\circ}))^{2} = (sec (33^{\circ}))^{2}- (tan (33^{\circ}))^{2}$

Recall the Pythagorean Identity:

$1 + (tan x)^{2} = (sec x)^{2}$

We can rearrange the terms:

$1 = (sec x)^{2} - (tan x)^{2}$

This is exactly what our original equation looks like, so the answer is 1.

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Question

Simplify $\frac{1}{1-\cos^2\theta}$ .

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Answer

To simplify, recognize that $1-\cos^2\theta$ is a reworking on $\sin^2+\cos^2=1$ , meaning that $1-\cos^2\theta=\sin^2\theta$ .

Plug that into our given equation:

$\frac{1}{1-\cos^2\theta}=\frac{1}{\sin^2\theta}$

Remember that $\csc\theta=\frac{1}{\sin\theta}$ , so $\frac{1}{\sin^2\theta}=\csc^2\theta$ .

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Question

Simplify $\frac{1-\cos^2\theta}{\sin\theta\cos\theta}$ .

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Answer

Recognize that $1-\cos^2\theta$ is a reworking on $\sin^2+\cos^2=1$ , meaning that $1-\cos^2\theta=\sin^2\theta$ .

Plug that in to our given equation:

$\frac{1-\cos^2\theta}{\sin\theta\cos\theta}=\frac{\sin^2\theta}{\sin\theta\cos\theta}$

Notice that one of the $\sin\theta$ 's cancel out.

$\frac{\sin\theta}{\cos\theta}=\tan\theta$ .

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Question

Simplify the expression: $(1+\cos a)(1-\cos a)$

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Answer

The expression $(1-\cos a)(1+\cos a)$ represents a difference of squares. In this case, the product is $1-\cos^{2}a$ (remember that 1 is also a perfect square).

One Pythagoran identity for trigonometric functions is:

$1-\cos^{2}a = \sin^{2}a$

Thus, we can say that the most simplified version of the expression is $\sin^{2}a$ .

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Question

Simplify the equation using identities:

$\cot^{2}x\cos x \sin x+ \cos x \sin x$

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Answer

There are a couple valid strategies for solving this problem. The simplest is to first factor out $\cos x\sin x$ from both sides. This leaves us with:

$\cot^{2}x\cos x\sin x + \cos x \sin x = (\cos x \sin x)(cot^{2}x + 1)$

Next, substitute with the known identity $\cot^{2}x + 1=\csc^{2}x$ to get:

$\cos x\sin x (\csc^{2}x)$

From here, we can eliminate the quadratic by converting:

$\csc^{2}x= \frac{1}{\sin^{2}x}$

giving us

$\frac{\cos x\sin x}{\sin^{2}x}=\frac{\cos x}{\sin x}= \cot x$

Thus,

$\cot^{2}x\cos x\sin x+\cos x\sin x = \cot x$

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Question

If theta is in the second quadrant, and $sin(\theta)= \frac{3}{5}$ , what is $cos(\theta)$ ?

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Answer

Write the Pythagorean Identity.

$sin^2(\theta)+cos^2(\theta)=1$

Substitute the value of $sin(\theta)$ and solve for $cos(\theta)$ .

$(\frac{3}{5})^2+cos^2(\theta)=1$

$cos^2(\theta)=1-(\frac{3}{5})^2$

$cos^2(\theta)=1-(\frac{3}{5})^2= \frac{25}{25}-\frac{9}{25}=\frac{16}{25}$

$cos(\theta)=\pm \sqrt{\frac{16}{25}}=\pm \frac{4}{5}$

Since the cosine is in the second quadrant, the correct answer is:

$cos(\theta)=-\frac{4}{5}$

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Question

$\sin^2x + \cos^2x + \tan^2x =$

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Answer

By the Pythagorean identity, the first two terms simplify to 1:

$\sin^2x+\cos^2x+\tan^2x=1+\tan^2x$ .

Dividing the Pythagorean identity by $\cos^2x$ allows us to simplify the right-hand side.

$\frac{\sin^2x}{\cos^2x}+\frac{\cos^2x}{\cos^2x}=\frac{1}{\cos^2x}$

$\tan^2x+1=\sec^2x$

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Question

For which values of is the following equation true?

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Answer

According to the Pythagorean identity

,

the right hand side of this equation can be rewritten as . This yields the equation

.

Dividing both sides by yields:

.

Dividing both sides by yields:

$tanx=\frac{sinx}{cosx}$ .

This is precisely the definition of the tangent function; since the domain of consists of all real numbers, the values of which satisfy the original equation also consist of all real numbers. Hence, the correct answer is

$\mathbb{R}=(-\infty, \infty )$ .

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Question

What is equal to?

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Answer

Step 1: Recall the trigonometric identity that has sine and cosine in it...

The sum is equal to 1.

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Question

Given $\sin(\theta)=\frac{1}{4}$ , what is $\cos(\theta)$ ?

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Answer

Using the Pythagorean Identity

$\sin^2+\cos^2=1$ ,

one can solve for $\cos^2(\theta)$ by plugging in for $\sin^2$ .

Solving for $\cos^2(\theta)$ , you get it equal to .

Taking the square root of both sides will get the correct answer of

$\cos(\theta)=\pm\sqrt{\frac{15}{4}}$ .

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Question

Reduce the following expression.

$\small \frac{\sin ^{2} x + \cos^{2} x}{\sin x}$

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Answer

Because $\sin ^{2} x + \cos ^{2} x = 1$ ,

therefore:

$\small \frac{\sin ^2 x + \cos^2 x}{\sin x} = \frac{1}{\sin x}$

By definition of cosecant,

$\small \frac{1}{\sin x} = \csc x$

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Question

Reduce the following expression.

$\small \tan ^2 x +1 - \sec^2 x$

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Answer

There are several ways to work this problem, but all of them use the second Pythagorean trig identity, $\small \tan ^2 x + 1 = \sec^2 x$ .

You can use this identity to substitute for parts of the expression. Here are two examples.

Method 1:

Substituting $-(\tan^2 x +1)$ for $\small -\sec^2 x$ , we get

$\tan ^2 x +1 - (\tan ^2 x +1)$ , which equals zero.

Method 2:

Substituting $\sec^2 x$ for $\tan^2 x +1$ , we get

$\sec^2 x-\sec^2 x$ , which equals zero.

Regardless of which substitution you choose, the answer is the same.

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Question

Simplify

$\frac{\sin^{3}{\left (x \right )} + \cos{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

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Answer

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{3}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} \sin{\left (x \right )} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

Now simplify, and combine each term.

$\\frac{1}{2} \sin{\left (x \right )} + \frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} \\\frac{1}{2} \left(\sin{\left (x \right )} + \cos{\left (x \right )} \csc^{2}{\left (x \right )}\right)$

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Question

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

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Answer

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

Now simplify, and combine each term.

$\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

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Question

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

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Answer

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

Now simplify, and combine each term.

$\\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

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Question

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

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Answer

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos^{2}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

Now simplify, and combine each term.

$\\frac{1}{2} \cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\frac{1}{2} \left(\cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

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Question

Simplify

$\frac{\sin^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

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Answer

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} \sin{\left (x \right )} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

Now simplify, and combine each term.

$\\frac{1}{2} \sin{\left (x \right )} + \frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} \\\frac{1}{2} \left(\sin{\left (x \right )} + \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}\right)$

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Question

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

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Answer

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

Now simplify, and combine each term.

$\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

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Question

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

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Answer

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

Now simplify, and combine each term.

$\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

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Question

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

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Answer

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$ Now simplify, and combine each term.

$\\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

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