Trigonometry Examples

$\csc^{2} (x) \tan^{2} (x) - 2 = \tan^{2} (x)$

Step 1

Replace the $\csc^{2} (x)$ with $\cot^{2} (x) + 1$ based on the $\cot^{2} (x) + 1 = \csc^{2} (x)$ identity.

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

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $\tan^{2} (x)$ by $1$ .

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

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $\cot^{2} (x) \tan^{2} (x) + \tan^{2} (x) - 2$ .

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 3.1.1.4

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3.1.1.5

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

Tap for more steps...

Step 3.1.1.5.1

Cancel the common factor.

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

Step 3.1.1.5.2

Rewrite the expression.

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

Step 3.1.1.6

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

Tap for more steps...

Step 3.1.1.6.1

Cancel the common factor.

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

Step 3.1.1.6.2

Rewrite the expression.

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

Step 3.1.2

Rearrange terms.

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

Step 3.1.3

Apply pythagorean identity.

$\sec^{2} (x) - 2 = \tan^{2} (x)$

Step 4

Replace the $\sec^{2} (x)$ with $1 + \tan^{2} (x)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

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

Step 5

Subtract $2$ from $1$ .

$\tan^{2} (x) - 1 = \tan^{2} (x)$

Step 6

Move all terms containing $\tan (x)$ to the left side of the equation.

Tap for more steps...

Step 6.1

Subtract $\tan^{2} (x)$ from both sides of the equation.

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

Step 6.2

Combine the opposite terms in $\tan^{2} (x) - 1 - \tan^{2} (x)$ .

Tap for more steps...

Step 6.2.1

Subtract $\tan^{2} (x)$ from $\tan^{2} (x)$ .

$0 - 1 = 0$

Step 6.2.2

Subtract $1$ from $0$ .

$- 1 = 0$

Step 7

Since $- 1 \neq 0$ , there are no solutions.

No solution