Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

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Step 2.1

Multiply by $1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Apply pythagorean identity.

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

Step 3

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

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

Step 4

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

$\sin (x)$

Step 5

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

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