Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

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

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

$1 - \sin (x) \sin (x)$

Step 2.6

Multiply $- \sin (x) \sin (x)$ .

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Step 2.6.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.6.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.6.4

Add $1$ and $1$ .

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

Step 2.7

Apply pythagorean identity.

$\cos^{2} (x)$

Step 3

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

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