Solve left(csc(θ)right)^2-left(cot(θ)right)^2=

$${ \left( \csc ( \theta ) \right) }^{ 2 } - { \left( \cot ( \theta ) \right) }^{ 2 } =$$

Factor

$$1$$

Solution Steps

The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(\frac{1}{\sin(\theta )}-\cot(\theta )\right)\left(\frac{1}{\sin(\theta )}+\cot(\theta )\right)$$

Consider \frac{1}{\sin(\theta )}-\cot(\theta ). Factor out \frac{1}{\sin(\theta )}.

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

Consider \frac{1}{\sin(\theta )}+\cot(\theta ). Factor out \frac{1}{\sin(\theta )}.

$$\frac{1+\cos(\theta )}{\sin(\theta )}$$

Rewrite the complete factored expression.

$$\left(1-\cos(\theta )\right)\left(1+\cos(\theta )\right)\times \left(\frac{1}{\sin(\theta )}\right)^{2}$$

Evaluate

$$1$$