Trigonometry Examples

$\frac{\tan^{2} (x + 1)}{\tan (x) \csc^{2} (x)} = \tan (x)$

Step 1

Subtract $\tan (x)$ from both sides of the equation.

$\frac{\tan^{2} (x + 1)}{\tan (x) \csc^{2} (x)} - \tan (x) = 0$

Step 2

Simplify each term.

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Step 2.1

Separate fractions.

$\frac{\tan^{2} (x + 1)}{\csc^{2} (x)} \cdot \frac{1}{\tan (x)} - \tan (x) = 0$

Step 2.2

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

$\frac{\tan^{2} (x + 1)}{\csc^{2} (x)} \cdot \frac{1}{\frac{\sin (x)}{\cos (x)}} - \tan (x) = 0$

Step 2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\tan^{2} (x + 1)}{\csc^{2} (x)} \cdot \frac{\cos (x)}{\sin (x)} - \tan (x) = 0$

Step 2.4

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

$\frac{\tan^{2} (x + 1)}{\csc^{2} (x)} \cot (x) - \tan (x) = 0$

Step 2.5

Rewrite $\frac{\tan^{2} (x + 1)}{\csc^{2} (x)}$ as ${(\frac{\tan (x + 1)}{\csc (x)})}^{2}$ .

${(\frac{\tan (x + 1)}{\csc (x)})}^{2} \cot (x) - \tan (x) = 0$

Step 2.6

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

${(\frac{\tan (x + 1)}{\frac{1}{\sin (x)}})}^{2} \cot (x) - \tan (x) = 0$

Step 2.7

Rewrite $\tan (x + 1)$ in terms of sines and cosines.

${(\frac{\frac{\sin (x + 1)}{\cos (x + 1)}}{\frac{1}{\sin (x)}})}^{2} \cot (x) - \tan (x) = 0$

Step 2.8

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

${(\frac{\sin (x + 1) \sin (x)}{\cos (x + 1)})}^{2} \cot (x) - \tan (x) = 0$

Step 2.9

Convert from $\frac{\sin (x + 1) \sin (x)}{\cos (x + 1)}$ to $\sin (x + 1) \tan (x + 1)$ .

${(\sin (x + 1) \tan (x + 1))}^{2} \cot (x) - \tan (x) = 0$

Step 2.10

Apply the product rule to $\sin (x + 1) \tan (x + 1)$ .

$\sin^{2} (x + 1) \tan^{2} (x + 1) \cot (x) - \tan (x) = 0$

Step 3

Set the argument in $\tan (x + 1)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x + 1 = \frac{π}{2} + π n$ , for any integer $n$

Step 4

Subtract $1$ from both sides of the equation.

$x = \frac{π}{2} + π n - 1$

Step 5

Set the argument in $\cot (x)$ equal to $π n$ to find where the expression is undefined.

$x = π n$ , for any integer $n$

Step 6

Set the argument in $\tan (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = \frac{π}{2} + π n - 1, x = π n, x = \frac{π}{2} + π n}$ $n$ , for any integer $n$

Step 8