Precalculus Examples

$(1 - \cos^{2} (θ)) (1 + \cot^{2} (θ)) = 1$

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

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Step 2.1

Apply pythagorean identity.

$\sin^{2} (θ) (1 + \cot^{2} (θ))$

Step 2.2

Apply pythagorean identity.

$\sin^{2} (θ) \csc^{2} (θ)$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

One to any power is one.

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

Step 4.2

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

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$1$

Step 5

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

$(1 - \cos^{2} (θ)) (1 + \cot^{2} (θ)) = 1$ is an identity