Question

prove that sin^4+Acos4^A=1-2sin^Acos^2A

Answer 1

We have,

=> sin⁴A + cos⁴A = 1 - 2sin²A.cos²A

Solving LHS :-

=> sin⁴A + cos⁴A

=> (sin²A)² + cos⁴A

=> (1 - cos²A)² + cos⁴A

{Since, sin²A = 1 - cos²A}

=> 1 + cos⁴A - 2cos²A + cos⁴A

=> 1 + 2cos⁴A - 2cos²A

=> 1 - 2cos²A(1 - cos²A)

=> 1 - 2cos²A.sin²A

{Since, sin²A = 1 - cos²A}

or, 1 - 2sin²A.cos²A

Q.E.D.