Simplify tan(theta)^2(cot(theta)^2-cos(theta)^2)

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

Step 1

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

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

Step 2

Simplify terms.

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

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

Simplify each term.

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

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} ({(\frac{\cos (θ)}{\sin (θ)})}^{2} - \cos^{2} (θ))$

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

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} (\frac{\cos^{2} (θ)}{\sin^{2} (θ)} - \cos^{2} (θ))$

Simplify terms.

Apply the distributive property.

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} \cdot \frac{\cos^{2} (θ)}{\sin^{2} (θ)} + \frac{\sin^{2} (θ)}{\cos^{2} (θ)} (- \cos^{2} (θ))$

Combine.

$\frac{\sin^{2} (θ) \cos^{2} (θ)}{\cos^{2} (θ) \sin^{2} (θ)} + \frac{\sin^{2} (θ)}{\cos^{2} (θ)} (- \cos^{2} (θ))$

Rewrite using the commutative property of multiplication.

$\frac{\sin^{2} (θ) \cos^{2} (θ)}{\cos^{2} (θ) \sin^{2} (θ)} - \frac{\sin^{2} (θ)}{\cos^{2} (θ)} \cos^{2} (θ)$

Simplify each term.

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

Cancel the common factor.

$\frac{\sin^{2} (θ) \cos^{2} (θ)}{\cos^{2} (θ) \sin^{2} (θ)} - \frac{\sin^{2} (θ)}{\cos^{2} (θ)} \cos^{2} (θ)$

Rewrite the expression.

$\frac{\cos^{2} (θ)}{\cos^{2} (θ)} - \frac{\sin^{2} (θ)}{\cos^{2} (θ)} \cos^{2} (θ)$

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

Cancel the common factor.

$\frac{\cos^{2} (θ)}{\cos^{2} (θ)} - \frac{\sin^{2} (θ)}{\cos^{2} (θ)} \cos^{2} (θ)$

Rewrite the expression.

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

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

Move the leading negative in $- \frac{\sin^{2} (θ)}{\cos^{2} (θ)}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

$1 - \sin^{2} (θ)$

Step 3

Apply pythagorean identity.

$\cos^{2} (θ)$