Trigonometry Examples

$\cos (x) (\sec (x) + \csc (x) \cot (x))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$\cos (x) (\frac{1}{\cos (x)} + \csc (x) \cot (x))$

Step 1.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$\cos (x) (\frac{1}{\cos (x)} + \frac{1}{\sin (x)} \cot (x))$

Step 1.3

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.4

Multiply $\frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}$ .

Tap for more steps...

Step 1.4.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

Add $1$ and $1$ .

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

Multiply $\cos (x) \frac{\cos (x)}{\sin^{2} (x)}$ .

Tap for more steps...

Step 3.1

Combine $\cos (x)$ and $\frac{\cos (x)}{\sin^{2} (x)}$ .

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

Step 3.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.3

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 4

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

$1 + \cot^{2} (x)$

Step 5

Apply pythagorean identity.

$\csc^{2} (x)$