Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Rewrite the expression.

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

Step 1.6

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 1.7

One to any power is one.

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

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $1$ and $1$ .

$\frac{2}{\sin^{2} (x)}$

Step 3

Multiply by $1$ .

$\frac{2}{\sin^{2} (x) \cdot 1}$

Step 4

Separate fractions.

$\frac{2}{1} \cdot \frac{1}{\sin^{2} (x)}$

Step 5

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

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

Step 6

Divide $2$ by $1$ .

$2 \csc^{2} (x)$