Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

Apply pythagorean identity.

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

Step 3

Rewrite $\frac{\cos^{2} (x)}{\cot^{2} (x)}$ as ${(\frac{\cos (x)}{\cot (x)})}^{2}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\sin^{2} (x)$