Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Multiply $\cos (θ)$ by $1$ .

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

Step 2.1.6

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

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

Step 2.1.7

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

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

Step 2.1.8

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

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

Step 2.1.9

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

$\cos^{2} (θ) + {(1 \sin (θ))}^{2}$

Step 2.1.10

Multiply $\sin (θ)$ by $1$ .

$\cos^{2} (θ) + \sin^{2} (θ)$

Step 2.2

Rearrange terms.

$\sin^{2} (θ) + \cos^{2} (θ)$

Step 2.3

Apply pythagorean identity.

$1$

Step 3

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{1}{\sec^{2} (θ)} + \frac{1}{\csc^{2} (θ)} = 1$ is an identity