Question

prove that sin^4 A - cos^4 A = 2 sin^2 A-1

Answer 1

To prove that :-

$\rm sin^4(A) - cos^4(A) = 2 sin^2(A) - 1$

Proof :-

$\rm sin^4(A) - cos^4(A) \\\\\implies (sin^2A - cos^2A)(sin^2A + cos^2A)\\\\\implies (sin^2A - cos^2A)\\\\\sf Since, a^2 - b^2 = (a + b)(a - b)\\and, sin^2A + cos^2A = 1\\\\So, (sin^2A- cos^2A)\\\\\implies (1 - 2cos^2A)\\\\\implies 1 - 2(1 - sin^2A)\\\\\implies 1 - 2 + 2sin^2A\\\\\implies 2sin^2A - 1$

Hence proved.

NOTE :-

$\sf sin^2A + cos^2 A = 1\\\\\implies sin^2 A + cos^2A - 2cos^2A = 1 - 2cos^2A\\\\\implies sin^2A - cos^2A = 1 - 2cos^2A\\ \\\\also, \\\\sin^2A + cos^2A = 1\\\\\implies cos^2A = 1 - sin^2A$

We have used these to prove the question.