Question

Simplify expression cos^2(pi/2-x) / √1-sin^2(x) =

Answer 1

Answer:

$\frac{cos^2(\frac{\pi}{2}-x)}{\sqrt{1-sin^2(x)}}=\frac{sin^2(x)}{|cos(x)|}$

Step-by-step explanation:

To simplify this expression you must use the following trigonometric identities

$cos(\frac{\pi}{2}-x) = sinx$     I

$1-sin (x) ^ 2 = cos ^ 2(x)$      II

Remember that

$\sqrt{f(x)^2} =f(x)$

Only if $f(x)> 0$  for all x

If f(x) is not greater than 0 for all x then

$\sqrt{f(x)^2} =|f(x)|$

Now we have the expression:

$\frac{cos^2(\frac{\pi}{2}-x)}{\sqrt{1-sin^2(x)}}$

then using the trigonometric identities I and II we have to:

$\frac{cos^2(\frac{\pi}{2}-x)}{\sqrt{1-sin^2(x)}}=\frac{sin^2(x)}{\sqrt{1-sin^2(x)}}\\\\\\\frac{sin^2(x)}{\sqrt{1-sin^2(x)}}= \frac{sin^2(x)}{\sqrt{cos^2(x)}}$

$cos(x)$ is not greater or equal than 0 for all x. So.

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

Finally  

$\frac{cos^2(\frac{\pi}{2}-x)}{\sqrt{1-sin^2(x)}}=\frac{sin^2(x)}{|cos(x)|}$