Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Convert from $\frac{\sin (y)}{\cos (y)}$ to $\tan (y)$ .

$\tan (y) = x$

Step 2.3

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

$y = \arctan (x)$

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

$f^{- 1} (\frac{\sec (x)}{\csc (x)}) = \arctan (\frac{\sec (x)}{\csc (x)})$

Step 4.2.3

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

$f^{- 1} (\frac{\sec (x)}{\csc (x)}) = \arctan (\frac{\sec (x)}{\frac{1}{\sin (x)}})$

Step 4.2.4

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

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

Step 4.2.5

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

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

Step 4.2.6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

$f (\arctan (x)) = \frac{\sec (\arctan (x))}{\csc (\arctan (x))}$

Step 4.3.3

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

$f (\arctan (x)) = \frac{\sec (\arctan (x))}{\frac{1}{\sin (\arctan (x))}}$

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

Convert from $\frac{\sin (\arctan (x))}{\cos (\arctan (x))}$ to $\tan (\arctan (x))$ .

$f (\arctan (x)) = \tan (\arctan (x))$

Step 4.3.7

The functions tangent and arctangent are inverses.

$f (\arctan (x)) = x$

Step 4.4

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

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