Question

Use trigonometric identities to simplify $sec^2(\pi /2-(x))[sin^2(x) -sin^4(x)]$

Answer 1

Answer:

Step-by-step explanation:

I am using trig identities and the formula for the difference of the cos of 2 angles to solve this. I'll do the steps one at a time. It's super tricky. First I'm just going to work on simplifying the sec² part and then I'll introduce the sin²(x) - sin⁴(x) when I need it. Beginning with the identity for the difference of the cos of 2 angles, knowing that sec²(x) = $\frac{1}{cos^2(x)}$:

$sec^2(\frac{\pi}{2}-x)=\frac{1}{cos^2(\frac{\pi}{2}-x )}$ and expand that using the formula for the difference:

$\frac{1}{cos(\frac{\pi}{2}-x)cos(\frac{\pi}{2}-x) }=$ $\frac{1}{(cos\frac{\pi}{2}cos(x)+sin\frac{\pi}{2}sin(x))(cos\frac{\pi}{2}cos(x)+sin\frac{\pi}{2}sin(x)) }$ and all of that simplifies down to

$\frac{1}{(0cos(x)+1sin(x))(0cos(x)+1sin(x))}$ which simplifies further to

$\frac{1}{(sin(x))(sin(x))}=\frac{1}{sin^2(x)}$ Now we'll bring in the other term. This is what we have now:

$\frac{1}{sin^2(x)}(\frac{sin^2(x)-sin^4(x)}{1})$ and distribute in to get:

$\frac{sin^2(x)}{sin^2(x)}-\frac{sin^4(x)}{sin^2(x)}$ which simplifies to

$1-sin^2(x)$ and that, finally, simplifies down to a simple

$cos^2(x)$