Trigonometry Examples

$\frac{\tan (x)}{\sec (x)}$

Step 1

Interchange the variables.

$x = \frac{\tan (y)}{\sec (y)}$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $\frac{\tan (y)}{\sec (y)} = x$ .

$\frac{\tan (y)}{\sec (y)} = x$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $\frac{\tan (y)}{\sec (y)}$ .

Tap for more steps...

Step 2.2.1.1

Rewrite $\sec (y)$ in terms of sines and cosines.

$\frac{\tan (y)}{\frac{1}{\cos (y)}} = x$

Step 2.2.1.2

Rewrite $\tan (y)$ in terms of sines and cosines.

$\frac{\frac{\sin (y)}{\cos (y)}}{\frac{1}{\cos (y)}} = x$

Step 2.2.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (y)}$ .

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

Step 2.2.1.4

Write $\cos (y)$ as a fraction with denominator $1$ .

$\frac{\sin (y)}{\cos (y)} \cdot \frac{\cos (y)}{1} = x$

Step 2.2.1.5

Cancel the common factor of $\cos (y)$ .

Tap for more steps...

Step 2.2.1.5.1

Cancel the common factor.

$\frac{\sin (y)}{\cos (y)} \cdot \frac{\cos (y)}{1} = x$

Step 2.2.1.5.2

Rewrite the expression.

$\sin (y) = x$

Step 2.3

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

$y = \arcsin (x)$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \arcsin (x)$

Step 4

Verify if $f^{- 1} (x) = \arcsin (x)$ is the inverse of $f (x) = \frac{\tan (x)}{\sec (x)}$ .

Tap for more steps...

Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

Tap for more steps...

Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\frac{\tan (x)}{\sec (x)})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\tan (x)}{\sec (x)}) = \arcsin (\frac{\tan (x)}{\sec (x)})$

Step 4.2.3

Rewrite $\sec (x)$ in terms of sines and cosines.

$f^{- 1} (\frac{\tan (x)}{\sec (x)}) = \arcsin (\frac{\tan (x)}{\frac{1}{\cos (x)}})$

Step 4.2.4

Rewrite $\tan (x)$ in terms of sines and cosines.

$f^{- 1} (\frac{\tan (x)}{\sec (x)}) = \arcsin (\frac{\frac{\sin (x)}{\cos (x)}}{\frac{1}{\cos (x)}})$

Step 4.2.5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$f^{- 1} (\frac{\tan (x)}{\sec (x)}) = \arcsin (\frac{\sin (x)}{\cos (x)} \cdot \cos (x))$

Step 4.2.6

Write $\cos (x)$ as a fraction with denominator $1$ .

$f^{- 1} (\frac{\tan (x)}{\sec (x)}) = \arcsin (\frac{\sin (x)}{\cos (x)} \cdot \frac{\cos (x)}{1})$

Step 4.2.7

Cancel the common factor of $\cos (x)$ .

Tap for more steps...

Step 4.2.7.1

Cancel the common factor.

$f^{- 1} (\frac{\tan (x)}{\sec (x)}) = \arcsin (\frac{\sin (x)}{\cos (x)} \cdot \frac{\cos (x)}{1})$

Step 4.2.7.2

Rewrite the expression.

$f^{- 1} (\frac{\tan (x)}{\sec (x)}) = \arcsin (\sin (x))$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

Tap for more steps...

Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (\arcsin (x))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arcsin (x)) = \frac{\tan (\arcsin (x))}{\sec (\arcsin (x))}$

Step 4.3.3

Rewrite $\sec (\arcsin (x))$ in terms of sines and cosines.

$f (\arcsin (x)) = \frac{\tan (\arcsin (x))}{\frac{1}{\cos (\arcsin (x))}}$

Step 4.3.4

Rewrite $\tan (\arcsin (x))$ in terms of sines and cosines.

$f (\arcsin (x)) = \frac{\frac{\sin (\arcsin (x))}{\cos (\arcsin (x))}}{\frac{1}{\cos (\arcsin (x))}}$

Step 4.3.5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (\arcsin (x))}$ .

$f (\arcsin (x)) = \frac{\sin (\arcsin (x))}{\cos (\arcsin (x))} \cdot \cos (\arcsin (x))$

Step 4.3.6

Write $\cos (\arcsin (x))$ as a fraction with denominator $1$ .

$f (\arcsin (x)) = \frac{\sin (\arcsin (x))}{\cos (\arcsin (x))} \cdot \frac{\cos (\arcsin (x))}{1}$

Step 4.3.7

Cancel the common factor of $\cos (\arcsin (x))$ .

Tap for more steps...

Step 4.3.7.1

Cancel the common factor.

$f (\arcsin (x)) = \frac{\sin (\arcsin (x))}{\cos (\arcsin (x))} \cdot \frac{\cos (\arcsin (x))}{1}$

Step 4.3.7.2

Rewrite the expression.

$f (\arcsin (x)) = \sin (\arcsin (x))$

Step 4.3.8

The functions sine and arcsine are inverses.

$f (\arcsin (x)) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arcsin (x)$ is the inverse of $f (x) = \frac{\tan (x)}{\sec (x)}$ .

$f^{- 1} (x) = \arcsin (x)$