Simplify (1-cos(x)^2)/(tan(x)^2)+2sin(x)^2

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

Step 1

Factor $1$ out of $\frac{1 - \cos^{2} (x)}{\tan^{2} (x)} + 2 \sin^{2} (x)$ .

Factor $1$ out of $\frac{1 - \cos^{2} (x)}{\tan^{2} (x)}$ .

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

Factor $1$ out of $2 \sin^{2} (x)$ .

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

Factor $1$ out of $1 \frac{1 - \cos^{2} (x)}{\tan^{2} (x)} + 1 (2 \sin^{2} (x))$ .

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

Step 2

Apply pythagorean identity.

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

Step 3

Simplify terms.

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

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

Simplify each term.

Rewrite $\frac{\sin^{2} (x)}{\tan^{2} (x)}$ as ${(\frac{\sin (x)}{\tan (x)})}^{2}$ .

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

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

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

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

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

Write $\sin (x)$ as a fraction with denominator $1$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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