Question

Part 1: Simplify each expression. Use the trigonometric identities to simplify each expression.

Answer 1

Answer:

$\cos ^2\left(x\right)$

Step-by-step explanation:

You will need to apply some basic trig identities here:

$\sec ^2\left(\frac{\pi }{2}-x\right) \\\\=> \frac{1}{\cos ^2\left(\frac{\pi }{2}-x\right)}\\\\=> \frac{1}{\sin ^2\left(x\right)}$

Now we substitute this value back into the given equation:

$\left(\frac{1}{\sin \left(x\right)}\right)^2\left(\sin ^2\left(x\right)-\sin ^4\left(x\right)\right)\\\\=> \frac{1\cdot \left(\sin ^2\left(x\right)-\sin ^4\left(x\right)\right)}{\sin ^2\left(x\right)}\\\\=> \frac{\sin ^2\left(x\right)-\sin ^4\left(x\right)}{\sin ^2\left(x\right)}\\\\=> \frac{\sin ^2\left(x\right)\left(1-\sin ^2\left(x\right)\right)}{\sin ^2\left(x\right)}\\\\=> 1-\sin ^2\left(x\right)\\\\=> \cos ^2\left(x\right)$

Hope that helps!