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Trigonometry : Pythagorean Identities

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Example Question #21 : Trigonometry

Simplify

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

Possible Answers:

$\frac{1}{2}$

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

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

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

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

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.

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

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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Example Question #22 : Trigonometry

Simplify

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

Possible Answers:

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

$\frac{1}{2}$

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

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

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

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 #23 : Trigonometry

Simplify

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

Possible Answers:

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

$\frac{1}{2}$

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

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

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

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.

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

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Example Question #24 : Trigonometry

Simplify

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

Possible Answers:

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

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

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

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

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

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

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Trigonometry : Pythagorean Identities

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #21 : Trigonometry

Example Question #22 : Trigonometry

Example Question #23 : Trigonometry

Example Question #24 : Trigonometry

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