Simplify csc(x)^2(1-sec(x)^2)

$\csc^{2} (x) (1 - \sec^{2} (x))$

Step 1

Simplify with factoring out.

Reorder $1$ and $- \sec^{2} (x)$ .

$\csc^{2} (x) (- \sec^{2} (x) + 1)$

Factor $- 1$ out of $- \sec^{2} (x)$ .

$\csc^{2} (x) (- (\sec^{2} (x)) + 1)$

Rewrite $1$ as $- 1 (- 1)$ .

$\csc^{2} (x) (- \sec^{2} (x) - 1 \cdot - 1)$

Factor $- 1$ out of $- \sec^{2} (x) - 1 \cdot - 1$ .

$\csc^{2} (x) (- (\sec^{2} (x) - 1))$

Step 2

Apply pythagorean identity.

$\csc^{2} (x) (- \tan^{2} (x))$

Step 3

Rewrite using the commutative property of multiplication.

$- \csc^{2} (x) \tan^{2} (x)$