Question

cos^(4) theta -sin^(4) theta = 2 cos^(2)theta-1 = 1-2 sin^(2)​

Answer 1

Given:  $y=$ $cos^{4}$$\alpha$$-sin^{4}\alpha$

To find: $y$$=cos^{4} \alpha - sin^{4} \alpha =2cos^{2} \alpha -1=1-2sin^{2} \alpha$, proof

Solution: $y=$ $cos^{4}$$\alpha$ $-sin^{4}\alpha$

                $y=(cos^{2}\alpha )^{2}$$- (sin^{2} \alpha )^{2}$

using formula $a^{2} -b^{2}$$=(a+b)(a-b)$    

          ⇒ $y=(cos^{2}\alpha + sin^{2} \alpha )(cos^{2}\alpha -sin^{2} \alpha )$

   since,    $sin^{2} \alpha +cos^{2}\alpha = 1$

            ⇒$y=cos^{2} \alpha -sin^{2} \alpha$

            ⇒ $y= 1-sin^{2} \alpha -sin^{2} \alpha$ , (using $cos^{2} \alpha =1-sin^{2} \alpha$)

            ⇒$y= 1-2sin^{2} \alpha$-------(1)

            ⇒$y=1-2(1-cos^{2} \alpha )$,   (using $sin^{2} \alpha =1-cos^{2} \alpha$)

            ⇒$y=2cos^{2} \alpha -1$--------(2)

          from equation (1) &(2)

            cos^(4) theta -sin^(4) theta = 2 cos^(2)theta-1 = 1-2 sin^(2)​

             hence proved