Find the Inverse 1/(csc(x)^2)+1/(sec(x)^2)

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Trigonometry

$\frac{1}{\csc^{2} (x)} + \frac{1}{\sec^{2} (x)}$

Simplify $\frac{1}{\csc^{2} (x)} + \frac{1}{\sec^{2} (x)}$ .

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Simplify each term.

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Rewrite $1$ as $1^{2}$ .

$y = \frac{1^{2}}{\csc^{2} (x)} + \frac{1}{\sec^{2} (x)}$

Rewrite $\frac{1^{2}}{\csc^{2} (x)}$ as ${(\frac{1}{\csc (x)})}^{2}$ .

$y = {(\frac{1}{\csc (x)})}^{2} + \frac{1}{\sec^{2} (x)}$

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

$y = {(\frac{1}{\frac{1}{\sin (x)}})}^{2} + \frac{1}{\sec^{2} (x)}$

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

$y = {(1 \sin (x))}^{2} + \frac{1}{\sec^{2} (x)}$

Multiply $\sin (x)$ by $1$ .

$y = \sin^{2} (x) + \frac{1}{\sec^{2} (x)}$

Rewrite $1$ as $1^{2}$ .

$y = \sin^{2} (x) + \frac{1^{2}}{\sec^{2} (x)}$

Rewrite $\frac{1^{2}}{\sec^{2} (x)}$ as ${(\frac{1}{\sec (x)})}^{2}$ .

$y = \sin^{2} (x) + {(\frac{1}{\sec (x)})}^{2}$

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

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

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

$y = \sin^{2} (x) + {(1 \cos (x))}^{2}$

Multiply $\cos (x)$ by $1$ .

$y = \sin^{2} (x) + \cos^{2} (x)$

Apply pythagorean identity.

$y = 1$

Interchange the variables.

$x = 1$

Since $y$ cannot be isolated, an inverse cannot be found.

Inverse function cannot be found

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