Trigonometry Examples

$\frac{1 + \cot (α)}{\csc (α)} = \sin (α) + \cos (α)$

Step 1

Start on the left side.

$\frac{1 + \cot (α)}{\csc (α)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Multiply $\sin (α)$ by $1$ .

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

Step 3.4

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

$\sin (α) + \cos (α)$

Step 4

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

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