Trigonometry Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

Apply pythagorean identity.

$\sin^{2} (x) (- \cot^{2} (x))$

Step 3

Rewrite using the commutative property of multiplication.

$- \sin^{2} (x) \cot^{2} (x)$

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

$- \cos^{2} (x)$