Precalculus Examples

$\cot^{2} (y) (\sec^{2} (y) - 1) = 1$

Step 1

Start on the left side.

$\cot^{2} (y) (\sec^{2} (y) - 1)$

Step 2

Apply pythagorean identity.

$\cot^{2} (y) \tan^{2} (y)$

Step 3

Convert to sines and cosines.

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Step 3.1

Write $\cot (y)$ in sines and cosines using the quotient identity.

${(\frac{\cos (y)}{\sin (y)})}^{2} \tan^{2} (y)$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Combine.

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

Step 4.2

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

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Cancel the common factor of ${\sin (y)}^{2}$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$1$

Step 5

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

$\cot^{2} (y) (\sec^{2} (y) - 1) = 1$ is an identity