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Trigonometry : Identities of Inverse Operations

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Example Question #1 : Identities Of Inverse Operations

Simplify using identities. Leave no fractions in your answer.

$\sec \theta(cos\theta\sin\theta) - \csc\theta\tan\theta$

Possible Answers:

$\sin\theta - \sec\theta$

$\cos\theta - \sec\theta$

$\sin\theta - \cos\theta$

$\cot\theta$

$\cos\theta-\sin\theta$

Correct answer:

$\sin\theta - \sec\theta$

Explanation:

The easiest first step is to simplify our inverse identities:

$\sec \theta(cos\theta\sin\theta) - \csc\theta\tan\theta = \bigg(\frac{1}{cos\theta}\bigg)cos\theta\sin\theta-\bigg(\frac{1}{\sin\theta}\bigg)\bigg(\frac{sin\theta}{\cos\theta}\bigg)$

Cross cancelling, we end up with

$\bigg(\frac{1}{cos\theta}\bigg)cos\theta\sin\theta-\bigg(\frac{1}{\sin\theta}\bigg)\bigg(\frac{\sin\theta}{\cos\theta}\bigg)=\sin\theta-\frac{1}{\cos\theta}$

Finally, eliminate the fraction:

$\sin\theta - \frac{1}{\cos\theta}= \sin\theta-\sec\theta$

Thus,

$\sec \theta(cos\theta\sin\theta) - \csc\theta\tan\theta = \sin\theta - \sec\theta$

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Example Question #1 : Identities Of Inverse Operations

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions.

1. $\frac{cot(x)}{tan(x)}$

Possible Answers:

cos^2(x)

cot^2(x)

tan^2(x)

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

Correct answer:

cot^2(x)

Explanation:

Using the quotient identities for trig functions, you can rewrite,

$cot(x)=\frac{cos(x)}{sin(x)}$

and

$tan(x)=\frac{sin(x)}{cos(x)}$

Then the fraction becomes

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

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Example Question #2 : Identities Of Inverse Operations

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions.

(sin^2(x)-1)(1-sin^2(x))

Possible Answers:

-cos^4(x)

1-sin^2(x)

cos^4(x)

-cos^2(x)(1-sin^2(x))

Correct answer:

-cos^4(x)

Explanation:

Use the Pythagorean Identities:

sin^2(x)-1=cos^2(x)

and

sin^2(x)-1=-cos^2(x)

Thus the expression becomes,

(-cos^2(x))(cos^2(x))=-cos^4(x) .

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Example Question #4 : Identities Of Inverse Operations

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions.

(cot(x) - csc(x))(cot(x)+csc(x))

Possible Answers:

sin^2(x)

cos^2(x)

Correct answer:

Explanation:

Use the distributive property (FOIL method) to simplify the expression.

(cot(x) - csc(x))(cot(x)+csc(x))

=cot^2(x)-cot(x)csc(x)+cot(x)csc(x)-csc^2(x)

=cot^2(x)-csc^2(x)

Using Pythagorean Identities:

=cot^2(x)-csc^2(x)=-1 .

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Example Question #5 : Identities Of Inverse Operations

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions.

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

Possible Answers:

2-sin(x)

-sin(x)

2+sin(x)

sin(x)

Correct answer:

-sin(x)

Explanation:

First, simplify the first term in the expression to 1 because of the Pythagorean Identity.

sin^2(x)+cos^2(x)=1

Then, simplify the second term to

$\frac{(1 - sin^2x)}{(1 - sinx)}$ .

This reduces to

$\frac{(1-sinx)(1+sinx)}{1-sinx}=1+sin(x)$ .

The expression is now,

1 - (1+sinx) .

Distribute the negative and get,

-sinx .

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Example Question #6 : Identities Of Inverse Operations

Solve each question over the interval $0 < x < 2\pi$

tanx = cotx

Possible Answers:

$\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$

$\frac{\pi}{2}, \frac{3\pi}{2}, 2\pi$

$\frac{\pi}{4}, \frac{5\pi}{4}$

$0, 2\pi$

Correct answer:

$\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$

Explanation:

Divide both sides by cotx to get tan^2x = 1 .

Take the square root of both sides to get that tanx = 1 and tanx = -1 .

The angles for which this is true (this is taking the arctan) are every angle when sinx = cosx and sinx = -cosx .

These angles are all the multiples of $\frac{\pi}{4}$ .

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Example Question #7 : Identities Of Inverse Operations

tan(x) can be stated as all of the following except...

Possible Answers:

tan(-x)

$cot(x)^{-1}$

$cot(\frac{\pi}{2} - x)$

$\frac{sin(x)}{cos(x)}$

-tan(-x)

Correct answer:

tan(-x)

Explanation:

Let's look at these individually:

$\frac{sin(x)}{cos(x)}$ is true by definition, as is $cot(x)^{-1}$ .

$cot(\frac{\pi}{2} - x)$ is also true because of a co-function identity.

This leaves two - and we can tell which of these does not work using the fact that tan(-x) = -tan(x) , which means that tan(-x) is our answer.

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Trigonometry : Identities of Inverse Operations

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Identities Of Inverse Operations

Example Question #1 : Identities Of Inverse Operations

Example Question #2 : Identities Of Inverse Operations

Example Question #4 : Identities Of Inverse Operations

Example Question #5 : Identities Of Inverse Operations

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