Question

sin⁴x-cos⁴x+1=2sin²x​

Answer 1

Answer:

To prove:

sin⁴x-cos⁴x+1 = 2sin²x​

Proof:

Solving LHS:

=sin⁴x-cos⁴x+1

= (sin² x)² - (cos² x)² +1                                        [ Identity: (a)²-(b)²= (a+b) (a-b) ]

= (sin² x + cos² x) × (sin² x - cos² x) + 1                [a= sin² x , b= cos² x]

= 1 × (sin² x - cos² x) + 1                                       [ sin² x + cos² x =1 ]

= sin² x - cos² x+ 1  

= sin² x - (1 - sin² x) + 1                          [ sin² x + cos² x =1 => cos² x = 1 - sin² x]

= sin² x - 1 + sin² x + 1                                [Opening the bracket]

= (sin² x + sin² x) + (1 - 1)

= 2sin² x = RHS

Hence, Proved