Trigonometry Examples

$\tan (x) (\csc (x) - \sin (x)) = \cos (x)$

Step 1

Start on the left side.

$\tan (x) (\csc (x) - \sin (x))$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

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

$\frac{\sin (x)}{\cos (x)} (\csc (x) - \sin (x))$

Step 2.2

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

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

Step 2.3

Apply the distributive property.

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

Step 2.4

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

Tap for more steps...

Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 2.5

Rewrite using the commutative property of multiplication.

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

Step 2.6

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

Tap for more steps...

Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

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

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

Step 2.6.5

Add $1$ and $1$ .

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Apply pythagorean identity.

$\frac{\cos^{2} (x)}{\cos (x)}$

Step 2.9

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

Tap for more steps...

Step 2.9.1

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

$\frac{\cos (x) \cos (x)}{\cos (x)}$

Step 2.9.2

Cancel the common factors.

Tap for more steps...

Step 2.9.2.1

Multiply by $1$ .

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

Step 2.9.2.2

Cancel the common factor.

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

Step 2.9.2.3

Rewrite the expression.

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

Step 2.9.2.4

Divide $\cos (x)$ by $1$ .

$\cos (x)$

Step 3

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

$\tan (x) (\csc (x) - \sin (x)) = \cos (x)$ is an identity