Trigonometry Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Factor $\csc (x)$ out of $- \csc (x)$ .

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

Step 1.1.3

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 2

Apply pythagorean identity.

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

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

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

Step 5

Cancel the common factor of $\sin (x)$ .

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Step 5.1

Move the leading negative in $- \frac{1}{\sin (x)}$ into the numerator.

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

Step 5.2

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

$\frac{- 1}{\sin (x)} (\sin (x) \sin (x))$

Step 5.3

Cancel the common factor.

$\frac{- 1}{\sin (x)} (\sin (x) \sin (x))$

Step 5.4

Rewrite the expression.

$- \sin (x)$