How do you verify csc^2(theta)(1-cos^2(theta))=1?

Nov 6, 2015

True

Explanation:

${\csc}^{2} \left(\theta\right) \cdot \left(1 - {\cos}^{2} \left(\theta\right)\right) = 1$

1) Solve on the left side. Notice the Pythagorean Identity.

$1 - {\cos}^{2} \left(\theta\right) = {\sin}^{2} \left(\theta\right)$, this is just a jumbled version of ${\sin}^{2} x + {\cos}^{2} x = 1$ identity.

2) implement the new value so that everything is sine on the left side.

$\frac{1}{\sin} ^ 2 \left(\theta\right) \cdot {\sin}^{2} \frac{\theta}{1} \to {\sin}^{2} \frac{\theta}{\sin} ^ 2 \left(\theta\right) \to 1$

It sounds like you just need to memorize the identities so you can spot them more easily in the future.