7 Verify the identities a cos x y sin x y sin x cos x

7. Verify the identities:

a. cos( x - y ) + sin( x + y ) =( sin x + cos x) (sin y + cos y)

b. (sin x + cos x)² = sin (2x) + 1

Solution

7a) cos( x - y ) + sin( x + y ) =( sin x + cos x) (sin y + cos y)

Use the formula for : cos(a - b) = cosacosb + sinasinb

                     and sin(a +b) - sinacosb +cosasinb

so, cos( x - y ) + sin( x + y ) = cosxcosy +sinxsiny + sinxcosy + cosxsiny

= cosxcosy+ sinxcosy + sinxsiny +cosxsiny

= cosy(cosx +sinx) +siny( sinx +cosx)

=(sinx+cosx)( cosy +siny)

= RHS

b) (sinx +cosx)^2 = sin2x +1

LHS : sin^2x +cos^2x +2sinxcosx

= 1 +sin2x                          { we know that as per trig identity: sin2x = 2sinxcosx}