Trigonometry Examples

$\cot (θ) + 1 = \csc (θ) (\cos (θ) + \sin (θ))$

Step 1

Start on the right side.

$\csc (θ) (\cos (θ) + \sin (θ))$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

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

$\frac{\cos (θ)}{\sin (θ)} + 1$

Step 4

Reorder terms.

$1 + \frac{\cos (θ)}{\sin (θ)}$

Step 5

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

$\cot (θ) + 1$

Step 6

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

$\cot (θ) + 1 = \csc (θ) (\cos (θ) + \sin (θ))$ is an identity