Trigonometry Examples

$\cot (x) - 1 = \cos (x) (\csc (x) - \sec (x))$

Step 1

Start on the right side.

$\cos (x) (\csc (x) - \sec (x))$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Combine $\cos (x)$ and $\frac{1}{\sin (x)}$ .

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

Step 3.3

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

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

Step 4

Reorder terms.

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

Step 5

Rewrite $- 1 + \frac{\cos (x)}{\sin (x)}$ as $\cot (x) - 1$ .

$\cot (x) - 1$

Step 6

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

$\cot (x) - 1 = \cos (x) (\csc (x) - \sec (x))$ is an identity