Trigonometry Examples

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

Step 1

Start on the left side.

$\sin^{2} (θ) \sec^{2} (θ)$

Step 2

Apply Pythagorean identity in reverse.

$(1 - \cos^{2} (θ)) \sec^{2} (θ)$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

One to any power is one.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply $\frac{1}{{\cos (θ)}^{2}}$ by $1$ .

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

Step 4.4

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

$\frac{1}{\cos^{2} (θ)} - 1$

Step 5

Reorder terms.

$- 1 + \frac{1}{\cos^{2} (θ)}$

Step 6

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

$\sec^{2} (θ) - 1$

Step 7

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

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