Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

Simplify.

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Step 1.8.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.8.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.8.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.8.4

Add $1$ and $1$ .

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

Step 2

Apply pythagorean identity.

$1$