Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 4

Simplify.

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Step 4.1

One to any power is one.

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

Step 4.2

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

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$1$

Step 5

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

$\csc^{2} (x) (1 - \cos^{2} (x)) = 1$ is an identity