Trigonometry Examples

$\frac{\sec^{2} (θ) - 1}{\sec^{2} (θ)} = \sin^{2} (θ)$

Step 1

Start on the left side.

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

Step 2

Apply pythagorean identity.

$\frac{\tan^{2} (θ)}{\sec^{2} (θ)}$

Step 3

Convert to sines and cosines.

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Step 3.1

Write $\tan (θ)$ in sines and cosines using the quotient identity.

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

Step 3.2

Apply the reciprocal identity to $\sec (θ)$ .

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

Step 3.3

Apply the product rule to $\frac{\sin (θ)}{\cos (θ)}$ .

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

Step 3.4

Apply the product rule to $\frac{1}{\cos (θ)}$ .

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Combine.

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

Step 4.3

Cancel the common factor of ${\cos (θ)}^{2}$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$\frac{{\sin (θ)}^{2}}{1^{2}}$

Step 4.4

One to any power is one.

$\frac{{\sin (θ)}^{2}}{1}$

Step 4.5

Divide ${\sin (θ)}^{2}$ by $1$ .

$\sin^{2} (θ)$

Step 5

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

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