Evaluate (tan(theta))/(sin(theta))=sec(theta)

$\frac{\tan (θ)}{\sin (θ)} = \sec (θ)$

Step 1

Simplify the left side.

Simplify $\frac{\tan (θ)}{\sin (θ)}$ .

Rewrite $\tan (θ)$ in terms of sines and cosines.

$\frac{\frac{\sin (θ)}{\cos (θ)}}{\sin (θ)} = \sec (θ)$

Rewrite $\frac{\frac{\sin (θ)}{\cos (θ)}}{\sin (θ)}$ as a product.

$\frac{\sin (θ)}{\cos (θ)} \cdot \frac{1}{\sin (θ)} = \sec (θ)$

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

Cancel the common factor.

$\frac{\sin (θ)}{\cos (θ)} \cdot \frac{1}{\sin (θ)} = \sec (θ)$

Rewrite the expression.

$\frac{1}{\cos (θ)} = \sec (θ)$

Step 2

Simplify the right side.

Rewrite $\sec (θ)$ in terms of sines and cosines.

$\frac{1}{\cos (θ)} = \frac{1}{\cos (θ)}$

Step 3

Multiply both sides of the equation by $\cos (θ)$ .

$\cos (θ) \frac{1}{\cos (θ)} = \cos (θ) \frac{1}{\cos (θ)}$

Step 4

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

Cancel the common factor.

$\cos (θ) \frac{1}{\cos (θ)} = \cos (θ) \frac{1}{\cos (θ)}$

Rewrite the expression.

$1 = \cos (θ) \frac{1}{\cos (θ)}$

Step 5

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

Cancel the common factor.

$1 = \cos (θ) \frac{1}{\cos (θ)}$

Rewrite the expression.

$1 = 1$

Step 6

Since $1 = 1$ , the equation will always be true.

Always true

Step 7

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$