Trigonometry Examples

$\frac{1}{1 - \sin^{2} (y)} = 1 + \tan^{2} (y)$

Step 1

Start on the right side.

$1 + \tan^{2} (y)$

Step 2

Simplify the expression.

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Step 2.1

Rearrange terms.

$\tan^{2} (y) + 1$

Step 2.2

Apply pythagorean identity.

$\sec^{2} (y)$

Step 2.3

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

${(\frac{1}{\cos (y)})}^{2}$

Step 2.4

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

$\frac{1^{2}}{\cos^{2} (y)}$

Step 2.5

One to any power is one.

$\frac{1}{\cos^{2} (y)}$

Step 3

Apply Pythagorean identity in reverse.

$\frac{1}{1 - \sin^{2} (y)}$

Step 4

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

$\frac{1}{1 - \sin^{2} (y)} = 1 + \tan^{2} (y)$ is an identity