Trigonometry Examples

$\cos (x) (\csc (x) + \tan (x)) = \cot (x) + \sin (x)$

Step 1

Start on the left side.

$\cos (x) (\csc (x) + \tan (x))$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Now consider the right side of the equation.

$\cot (x) + \sin (x)$

Step 5

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 6

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

$\cos (x) (\csc (x) + \tan (x)) = \cot (x) + \sin (x)$ is an identity