Simplify (1-sin(theta)^2)(1+cot(theta)^2)

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

Step 1

Apply pythagorean identity.

$\cos^{2} (θ) (1 + \cot^{2} (θ))$

Step 2

Apply pythagorean identity.

$\cos^{2} (θ) \csc^{2} (θ)$

Step 3

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

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

Step 4

Simplify the expression.

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

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

One to any power is one.

$\cos^{2} (θ) \frac{1}{\sin^{2} (θ)}$

Step 5

Combine $\cos^{2} (θ)$ and $\frac{1}{\sin^{2} (θ)}$ .

$\frac{\cos^{2} (θ)}{\sin^{2} (θ)}$

Step 6

Convert from $\frac{\cos^{2} (θ)}{\sin^{2} (θ)}$ to $\cot^{2} (θ)$ .

$\cot^{2} (θ)$