Determine if True cot(theta)+tan(theta)=sec(theta)csc(theta)

$\cot (θ) + \tan (θ) = \sec (θ) \csc (θ)$

Step 1

Start on the left side.

$\cot (θ) + \tan (θ)$

Step 2

Convert to sines and cosines.

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

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

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

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

Step 3

Add fractions.

To write $\frac{\cos (θ)}{\sin (θ)}$ as a fraction with a common denominator, multiply by $\frac{\cos (θ)}{\cos (θ)}$ .

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

To write $\frac{\sin (θ)}{\cos (θ)}$ as a fraction with a common denominator, multiply by $\frac{\sin (θ)}{\sin (θ)}$ .

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

Write each expression with a common denominator of $\sin (θ) \cos (θ)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $\sin (θ) \cos (θ)$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

$\frac{\cos^{2} (θ) + \sin^{2} (θ)}{\cos (θ) \sin (θ)}$

Step 5

Apply Pythagorean identity.

Rearrange terms.

$\frac{\sin^{2} (θ) + \cos^{2} (θ)}{\cos (θ) \sin (θ)}$

Apply pythagorean identity.

$\frac{1}{\cos (θ) \sin (θ)}$

Step 6

Rewrite $\frac{1}{\cos (θ) \sin (θ)}$ as $\sec (θ) \csc (θ)$ .

$\sec (θ) \csc (θ)$

Step 7

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

$\cot (θ) + \tan (θ) = \sec (θ) \csc (θ)$ is an identity