1.

Prove that sin ^4 theta -cos^4 theta =1-2cos^2thet​

Answer»

To prove :

sin⁴ theta - cos⁴ theta = 1 - 2 cos² theta

Solution :

LHS -

sin⁴ theta - cos⁴ theta

=> [ sin² theta ] ² - [ cos² theta ] ²

We know that , a² - b³ can be written as (a + b)( a - b)

Utilising that identity here ;

=> [ sin² theta + cos ² theta ][ sin² theta - cos² theta]

Now , sin² theta + cos ² theta = 1

=> [ 1 ] [ sin² theta - cos ² theta ]

sin² theta + cos ² theta = 1

=> sin² theta = 1 - cos² theta

=> ( 1 - cos² theta ) - cos² theta

=> [ 1 - 2 cos ² theta ]

So ,

sin⁴ theta - cos⁴ theta = 1 - 2sin² theta = 1 - 2 cos² theta

Hence PROVED

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