Determine if True cot(theta)sin(theta)cos(theta)=1-sin(theta)^2

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

Start on the left side.

$\cot (θ) \sin (θ) \cos (θ)$

Simplify the expression.

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

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

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

Cancel the common factors.

$\cos (θ) \cos (θ)$

Multiply $\cos (θ) \cos (θ)$ .

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

$\cos^{1} (θ) \cos (θ)$

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

$\cos^{1} (θ) \cos^{1} (θ)$

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

${\cos (θ)}^{1 + 1}$

Add $1$ and $1$ .

$\cos^{2} (θ)$

Apply Pythagorean identity in reverse.

$1 - \sin^{2} (θ)$

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