Trigonometry Examples

$\tan (x) + \frac{1}{\csc (x)} = \frac{1 + \cos (x)}{\cot (x)}$

Step 1

Start on the left side.

$\tan (x) + \frac{1}{\csc (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify each term.

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

Step 4

Now consider the right side of the equation.

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $\frac{\sin (x)}{\cos (x)}$ by $1$ .

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

Step 6.4

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

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

Step 7

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

$\tan (x) + \frac{1}{\csc (x)} = \frac{1 + \cos (x)}{\cot (x)}$ is an identity