Trigonometry Examples

$5 \sin^{2} (x) \csc^{2} (x) - 5 \sin^{2} (x)$

Step 1

Reorder $5 \sin^{2} (x) \csc^{2} (x)$ and $- 5 \sin^{2} (x)$ .

$- 5 \sin^{2} (x) + 5 \sin^{2} (x) \csc^{2} (x)$

Step 2

Factor $5 \sin^{2} (x)$ out of $- 5 \sin^{2} (x)$ .

$5 \sin^{2} (x) (- 1) + 5 \sin^{2} (x) \csc^{2} (x)$

Step 3

Factor $5 \sin^{2} (x)$ out of $5 \sin^{2} (x) \csc^{2} (x)$ .

$5 \sin^{2} (x) (- 1) + 5 \sin^{2} (x) (\csc^{2} (x))$

Step 4

Factor $5 \sin^{2} (x)$ out of $5 \sin^{2} (x) (- 1) + 5 \sin^{2} (x) (\csc^{2} (x))$ .

$5 \sin^{2} (x) (- 1 + \csc^{2} (x))$

Step 5

Rearrange terms.

$5 \sin^{2} (x) (\csc^{2} (x) - 1)$

Step 6

Apply pythagorean identity.

$5 \sin^{2} (x) \cot^{2} (x)$

Step 7

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

$5 \sin^{2} (x) {(\frac{\cos (x)}{\sin (x)})}^{2}$

Step 8

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

$5 \sin^{2} (x) \frac{\cos^{2} (x)}{\sin^{2} (x)}$

Step 9

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

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Step 9.1

Factor $\sin^{2} (x)$ out of $5 \sin^{2} (x)$ .

$\sin^{2} (x) \cdot 5 \frac{\cos^{2} (x)}{\sin^{2} (x)}$

Step 9.2

Cancel the common factor.

$\sin^{2} (x) \cdot 5 \frac{\cos^{2} (x)}{\sin^{2} (x)}$

Step 9.3

Rewrite the expression.

$5 \cos^{2} (x)$