Identities of Inverse Operations - Trigonometry

Card 0 of 28

Simplify using identities. Leave no fractions in your answer.

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

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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,

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

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

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

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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

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

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Use the Pythagorean Identities:

and

Thus the expression becomes,

.

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

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

Using Pythagorean Identities:

.

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

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

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,

.

Distribute the negative and get,

.

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

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Divide both sides by to get .

Take the square root of both sides to get that and .

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

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

can be stated as all of the following except...

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Let's look at these individually:

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

$cot($\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 , which means that is our answer.

Simplify using identities. Leave no fractions in your answer.

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

Tap to see back →

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,

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

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

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

Tap to see back →

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

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

Tap to see back →

Use the Pythagorean Identities:

and

Thus the expression becomes,

.

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

Tap to see back →

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

Using Pythagorean Identities:

.

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

Tap to see back →

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

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,

.

Distribute the negative and get,

.

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

Tap to see back →

Divide both sides by to get .

Take the square root of both sides to get that and .

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

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

can be stated as all of the following except...

Tap to see back →

Let's look at these individually:

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

$cot($\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 , which means that is our answer.

Simplify using identities. Leave no fractions in your answer.

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

Tap to see back →

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,

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

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

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

Tap to see back →

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

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

Tap to see back →

Use the Pythagorean Identities:

and

Thus the expression becomes,

.

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

Tap to see back →

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

Using Pythagorean Identities:

.

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

Tap to see back →

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

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,

.

Distribute the negative and get,

.

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

Tap to see back →

Divide both sides by to get .

Take the square root of both sides to get that and .

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

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