Question

Prove that
(sin^4 theta+cos^4 theta)
____________________=1
1-2sin^2 theta cos^2theta

Answer 1

Let your "theta"be denoted by "x".
So, your question is sin^4 x + cos^4 x /(1-2sin^2 x cos^2 x) =1.
Solution is as follows :
Consider the expression
sin^4 x + cos^4 x = (sin^4 x + 2sin^2 x cos^2 x + cos^4) - 2sin^2 x cos^2 x
=(sin^2 x + cos^2 x)^2 -2sin^2 x cos^2
= 1 - 2sin^2 x cos^2
Now,
sin^4 x + cos^4 x = 1 - 2sin^2 x cos^2
=> sin^4 x + cos^4 x/(1-2sin^2 x cos^2 x) =1.
Happy to help you :)