Trigonometry Examples

$\cot (θ) + \csc (θ) = \frac{\sin (θ) + \tan (θ)}{\sin (θ) \tan (θ)}$

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Factor $\sin (θ)$ out of $\sin (θ) + \frac{\sin (θ)}{\cos (θ)}$ .

Tap for more steps...

Step 3.1.1.1

Multiply by $1$ .

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

Step 3.1.1.2

Factor $\sin (θ)$ out of $\frac{\sin (θ)}{\cos (θ)}$ .

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

Step 3.1.1.3

Factor $\sin (θ)$ out of $\sin (θ) \cdot 1 + \sin (θ) \frac{1}{\cos (θ)}$ .

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

Step 3.1.2

Write $1$ as a fraction with a common denominator.

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

Step 3.1.3

Combine the numerators over the common denominator.

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

Step 3.2

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

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

Step 3.3

Simplify the numerator.

Tap for more steps...

Step 3.3.1

Raise $\sin (θ)$ to the power of $1$ .

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

Step 3.3.2

Raise $\sin (θ)$ to the power of $1$ .

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

Step 3.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.4

Add $1$ and $1$ .

$\frac{\sin (θ) \frac{\cos (θ) + 1}{\cos (θ)}}{\frac{{\sin (θ)}^{2}}{\cos (θ)}}$

Step 3.4

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

$\frac{\frac{\sin (θ) (\cos (θ) + 1)}{\cos (θ)}}{\frac{{\sin (θ)}^{2}}{\cos (θ)}}$

Step 3.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sin (θ) (\cos (θ) + 1)}{\cos (θ)} \cdot \frac{\cos (θ)}{{\sin (θ)}^{2}}$

Step 3.6

Combine.

$\frac{\sin (θ) (\cos (θ) + 1) \cos (θ)}{\cos (θ) {\sin (θ)}^{2}}$

Step 3.7

Cancel the common factor of $\sin (θ)$ and ${\sin (θ)}^{2}$ .

Tap for more steps...

Step 3.7.1

Factor $\sin (θ)$ out of $\sin (θ) (\cos (θ) + 1) \cos (θ)$ .

$\frac{\sin (θ) ((\cos (θ) + 1) \cos (θ))}{\cos (θ) {\sin (θ)}^{2}}$

Step 3.7.2

Cancel the common factors.

Tap for more steps...

Step 3.7.2.1

Factor $\sin (θ)$ out of $\cos (θ) {\sin (θ)}^{2}$ .

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

Step 3.7.2.2

Cancel the common factor.

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

Step 3.7.2.3

Rewrite the expression.

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

Step 3.8

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

$\frac{\cos (θ) + 1}{\sin (θ)}$

Step 4

Now consider the left side of the equation.

$\cot (θ) + \csc (θ)$

Step 5

Convert to sines and cosines.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 6

Combine the numerators over the common denominator.

$\frac{\cos (θ) + 1}{\sin (θ)}$

Step 7

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

$\cot (θ) + \csc (θ) = \frac{\sin (θ) + \tan (θ)}{\sin (θ) \tan (θ)}$ is an identity