Trigonometry Examples

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

Step 1

Start on the left side.

$\cot (θ) \sin (θ) \cos (θ)$

Step 2

Simplify the expression.

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Step 2.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 2.1.1

Rewrite $\cot (θ) \sin (θ)$ in terms of sines and cosines.

$\frac{\cos (θ)}{\sin (θ)} \sin (θ) \cos (θ)$

Step 2.1.2

Cancel the common factors.

$\cos (θ) \cos (θ)$

Step 2.2

Multiply $\cos (θ) \cos (θ)$ .

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Step 2.2.1

Raise $\cos (θ)$ to the power of $1$ .

$\cos^{1} (θ) \cos (θ)$

Step 2.2.2

Raise $\cos (θ)$ to the power of $1$ .

$\cos^{1} (θ) \cos^{1} (θ)$

Step 2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\cos (θ)}^{1 + 1}$

Step 2.2.4

Add $1$ and $1$ .

$\cos^{2} (θ)$

Step 3

Apply Pythagorean identity in reverse.

$1 - \sin^{2} (θ)$

Step 4

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

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