Expert: Ahmed Salami - 1/8/2005

Question
cos^4A- sin^4= cos2A prove this with steps


Answer
Hi Mukhtar,
Please forgive the delay. I was on vacation.
Firstly, we know that
cos(A+B) = cosAcosB - sinAsinB
therefore,
cos2A = cos(A+A) = cos^2A - sin^2A
In algebra, we also know that
a^2 - b^2 = (a+b)(a-b)
So
cos^4A - sin^4A = (cos^2A + sin^2A).(cos^2A - sin^2A)
but cos^2A + sin^2A = 1 (an identity)
We therefore have,
cos^4A - sin^4A = cos^2A - sin^2A
I hope you get it.
Regards.