Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

Cancel the common factor.

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

Step 4.1.1.2

Rewrite the expression.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Multiply $- 1$ by $1$ .

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

Step 4.1.4

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

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Step 4.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.4.2

Multiply ${\cos (x)}^{2}$ by $1$ .

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

Step 4.2

Subtract $1$ from $1$ .

$0 + {\cos (x)}^{2}$

Step 4.3

Add $0$ and ${\cos (x)}^{2}$ .

$\cos^{2} (x)$

Step 5

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

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