Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

Apply the reciprocal identity to $\csc (x)$ .

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

Step 3.2

Apply the reciprocal identity to $\csc (x)$ .

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Multiply $\frac{1}{\sin (x)}$ by $1$ .

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

Cancel the common factor.

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

Step 4.1.3.3

Rewrite the expression.

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

Subtract $1$ from $1$ .

$- \sin (x) + \frac{0}{\sin (x)}$

Step 4.4

Divide $0$ by $\sin (x)$ .

$- \sin (x) + 0$

Step 4.5

Add $- \sin (x)$ and $0$ .

$- \sin (x)$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cos^{2} (x) \csc (x) - \csc (x) = - \sin (x)$ is an identity