Trigonometry Examples

$\tan (- x) \csc (x)$

Step 1

Interchange the variables.

$x = \tan (- y) \csc (y)$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $\tan (- y) \csc (y) = x$ .

$\tan (- y) \csc (y) = x$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $\tan (- y) \csc (y)$ .

Tap for more steps...

Step 2.2.1.1

Since $\tan (- y)$ is an odd function, rewrite $\tan (- y)$ as $- \tan (y)$ .

$- \tan (y) \csc (y) = x$

Step 2.2.1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

Tap for more steps...

Step 2.2.1.2.1

Add parentheses.

$- (\tan (y) \csc (y)) = x$

Step 2.2.1.2.2

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

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

Step 2.2.1.2.3

Cancel the common factors.

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

Step 2.2.1.3

Convert from $\frac{1}{\cos (y)}$ to $\sec (y)$ .

$- \sec (y) = x$

Step 2.3

Divide each term in $- \sec (y) = x$ by $- 1$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $- \sec (y) = x$ by $- 1$ .

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

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2

Divide $\sec (y)$ by $1$ .

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

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$\sec (y) = - 1 \cdot x$

Step 2.3.3.2

Rewrite $- 1 \cdot x$ as $- x$ .

$\sec (y) = - x$

Step 2.4

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

$y = arcsec (- x)$

Step 3

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

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

Step 4

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

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

$f^{- 1} (\tan (- x) \csc (x)) = arcsec (- (\tan (- x) \csc (x)))$

Step 4.2.3

Since $\tan (- x)$ is an odd function, rewrite $\tan (- x)$ as $- \tan (x)$ .

$f^{- 1} (\tan (- x) \csc (x)) = arcsec (- (- \tan (x) \csc (x)))$

Step 4.2.4

Rewrite in terms of sines and cosines, then cancel the common factors.

Tap for more steps...

Step 4.2.4.1

Add parentheses.

$f^{- 1} (\tan (- x) \csc (x)) = arcsec (\tan (x) \csc (x))$

Step 4.2.4.2

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

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

Step 4.2.4.3

Cancel the common factors.

$f^{- 1} (\tan (- x) \csc (x)) = arcsec (\frac{1}{\cos (x)})$

Step 4.2.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

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

$f (arcsec (- x)) = \tan (- (arcsec (- x))) \csc (arcsec (- x))$

Step 4.3.3

Since $\tan (- arcsec (- x))$ is an odd function, rewrite $\tan (- arcsec (- x))$ as $- \tan (arcsec (- x))$ .

$f (arcsec (- x)) = - \tan (arcsec (- x)) \csc (arcsec (- x))$

Step 4.3.4

Rewrite in terms of sines and cosines, then cancel the common factors.

Tap for more steps...

Step 4.3.4.1

Add parentheses.

$f (arcsec (- x)) = - (\tan (arcsec (- x)) \csc (arcsec (- x)))$

Step 4.3.4.2

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

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

Step 4.3.4.3

Cancel the common factors.

$f (arcsec (- x)) = - \frac{1}{\cos (arcsec (- x))}$

Step 4.3.5

Convert from $\frac{1}{\cos (arcsec (- x))}$ to $\sec (arcsec (- x))$ .

$f (arcsec (- x)) = - \sec (arcsec (- x))$

Step 4.3.6

The functions secant and arcsecant are inverses.

$f (arcsec (- x)) = x$

Step 4.4

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

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