Trigonometry Examples

$(\sin (x)) (\cot (x) + \cos (x) \tan (x)) = \cos (x) + \sin^{2} (x)$

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\cos (x) + \sin (x) \sin (x)$

Step 3.4

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

$\cos (x) + \sin^{2} (x)$

Step 4

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

$(\sin (x)) (\cot (x) + \cos (x) \tan (x)) = \cos (x) + \sin^{2} (x)$ is an identity