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High School Math : Using Pythagorean Identities

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Example Questions

Example Question #1 : Using Pythagorean Identities

Simplify

$\cot(x)\times{\sin(x) }$ $\displaystyle \cot(x)\times{\sin(x) }$

Possible Answers:

$\arcsin(\pi )$ $\displaystyle \arcsin(\pi )$

$\arcsin(x)$ $\displaystyle \arcsin(x)$

$\cos(x)$ $\displaystyle \cos(x)$

$\csc(x)$ $\displaystyle \csc(x)$

$\tan(x)$ $\displaystyle \tan(x)$

Correct answer:

$\cos(x)$ $\displaystyle \cos(x)$

Explanation:

$\cot(x) = \cos(x)/\sin(x)$ $\displaystyle \cot(x) = \cos(x)/\sin(x)$. Thus: $[\cos(x)/\sin(x)]\times\sin(x)=\cos(x)$ $\displaystyle [\cos(x)/\sin(x)]\times\sin(x)=\cos(x)$

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Example Question #2 : Using Pythagorean Identities

Simplify

$\frac{cot (x)\cdot tan(x)}{cos(x)}$ $\displaystyle \frac{cot (x)\cdot tan(x)}{cos(x)}$

Possible Answers:

$\cos(x)$ $\displaystyle \cos(x)$

$\sec(x)$ $\displaystyle \sec(x)$

$\cot(x)/sec(x)$ $\displaystyle \cot(x)/sec(x)$

$\csc(x)$ $\displaystyle \csc(x)$

$\cos(x)\times\tan(x)$ $\displaystyle \cos(x)\times\tan(x)$

Correct answer:

$\sec(x)$ $\displaystyle \sec(x)$

Explanation:

$\cot(x)\times\tan(x) = 1$ $\displaystyle \cot(x)\times\tan(x) = 1$

and

$1/(\cos(x)) = \sec(x)$ $\displaystyle 1/(\cos(x)) = \sec(x)$.

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Example Question #6 : Trigonometric Identities

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

Possible Answers:

$-\sin x$ $\displaystyle -\sin x$

$\sin^2 x$ $\displaystyle \sin^2 x$

$\sin x$ $\displaystyle \sin x$

$\sin x\cos x$ $\displaystyle \sin x\cos x$

$\tan x$ $\displaystyle \tan x$

Correct answer:

$-\sin x$ $\displaystyle -\sin x$

Explanation:

Remember that $\sin^2x+\cos^2 x=1$ $\displaystyle \sin^2x+\cos^2 x=1$. We can rearrange this to simplify our given equation:

$\frac{\cos^2x-1}{\sin x}$ $\displaystyle \frac{\cos^2x-1}{\sin x}$

$=\frac{-\sin^2x}{\sin x}=-\sin x$ $\displaystyle =\frac{-\sin^2x}{\sin x}=-\sin x$

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Example Question #1 : Trigonometric Identities

Simplify:

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

Possible Answers:

$\csc^2 x$ $\displaystyle \csc^2 x$

$\sin x\cos x$ $\displaystyle \sin x\cos x$

$\sec^2x$ $\displaystyle \sec^2x$

$2\cos x+\cot x$ $\displaystyle 2\cos x+\cot x$

This is the most simplified version.

Correct answer:

$\sec^2x$ $\displaystyle \sec^2x$

Explanation:

Whenever you see a trigonometric function squared, start looking for a Pythagorean identity.

The two identities used in this problem are $\sin^2 x + \cos ^2 x=1$ $\displaystyle \sin^2 x + \cos ^2 x=1$ and $\tan^2 x +1 =\sec^2 x$ $\displaystyle \tan^2 x +1 =\sec^2 x$.

Substitute and solve.

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

$(\sin^2x+\cos^2x)+\tan^2x=?$ $\displaystyle (\sin^2x+\cos^2x)+\tan^2x=?$

$(1)+\tan^2x=\sec^2 x$ $\displaystyle (1)+\tan^2x=\sec^2 x$

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High School Math : Using Pythagorean Identities

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Using Pythagorean Identities

Example Question #2 : Using Pythagorean Identities

Example Question #6 : Trigonometric Identities

Example Question #1 : Trigonometric Identities

All High School Math Resources

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