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Trigonometry : Pythagorean Identities

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Example Question #11 : Trigonometry

Reduce the following expression.

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

Possible Answers:

$1+\cot ^{2} x$

$\sec x$

$\csc x$

Correct answer:

$\csc x$

Explanation:

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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Example Question #11 : Trigonometric Identities

Reduce the following expression.

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

Possible Answers:

$\small 2\tan ^2 x$

$\small -\sec x$

$\small \sec ^2 x - 1$

Correct answer:

Explanation:

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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Example Question #13 : Trigonometry

Simplify

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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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Example Question #14 : Trigonometry

Simplify

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

Possible Answers:

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

$\frac{1}{2}$

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

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

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

Correct answer:

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

Explanation:

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.

Now simplify, and combine each term.

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

Report an Error

Example Question #15 : Trigonometry

Simplify

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

Possible Answers:

$\frac{1}{2}$

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

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

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

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

Correct answer:

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

Explanation:

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.

Now simplify, and combine each term.

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

Report an Error

Example Question #16 : Trigonometry

Simplify

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

Possible Answers:

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

$\frac{1}{2}$

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

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

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

Correct answer:

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

Explanation:

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.

Now simplify, and combine each term.

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

Report an Error

Example Question #11 : Trigonometry

Simplify

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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.

Now simplify, and combine each term.

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

Report an Error

Example Question #18 : Trigonometry

Simplify

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

Possible Answers:

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

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

$\frac{1}{2}$

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

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

Correct answer:

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

Explanation:

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.

Now simplify, and combine each term.

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

Report an Error

Example Question #19 : Trigonometry

Simplify

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

Possible Answers:

$\frac{1}{2}$

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

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

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

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

Correct answer:

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

Explanation:

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.

Now simplify, and combine each term.

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

Report an Error

Example Question #20 : Trigonometry

Simplify

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

Possible Answers:

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

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

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

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

$\frac{1}{2}$

Correct answer:

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

Explanation:

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.

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

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Trigonometry : Pythagorean Identities

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #11 : Trigonometry

Example Question #11 : Trigonometric Identities

Example Question #13 : Trigonometry

Example Question #14 : Trigonometry

Example Question #15 : Trigonometry

Example Question #16 : Trigonometry

Example Question #11 : Trigonometry

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