Question

sin square alpha plus sin square beta = 1​

Answer 1

Answer:

Step-by-step explanation:

cos4αcos2β+sin4αsin2β=1

⇒cos4αsin2β+sin4αcos2β=cos2βsin2β

⇒cos4α(1−cos2β)+sin4αcos2β=cos2β(1−cos2β)

⇒cos4α−cos4αcos2β+sin4αcos2β=cos2β−cos4β

⇒cos4α−cos4αcos2β+(1−cos2α)2cos2β=cos2β−cos4β

⇒cos4α−cos4αcos2β+(1+cos4α−2cos2α)2cos2β=cos2β−cos4β

⇒cos4α−cos4αcos2β+cos2β+cos2βcos4α−2cos2αcos2β=cos2β−cos4β

⇒2cos4α=2cos2αcos2β

⇒cos2α=cos2β→(1)

⇒1−sin2α=1−sin2β

⇒sin2α=sin2β→(2)

Now,

(i)L.H.S.=sin4α+sin4β=(sin2α−sinβ)2+2sin2αsin2β

As, sin2α=sin2β,above expression becomes,

=0+2sin2αsin2β=2sin2αsin2β=R.H.S.

(ii)L.H.S.=cos4βcos2α+sin4βsin2α

From (1),

=cos4αcos2α+sin4αsin2α

=cosα+sin2α=1=R.H.S.=