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Question

If sin α+sin β=a and cos α+cos β=b, prove that

(i)  sin(α+β)=2aba2+b2

(ii)  cos(αβ)=a2+b222


Solution

(i) We have,

sin α+sin β=a and cos α+cos β=b     . . .(A)

Squaring and adding, we get

sin2α+sin2β+2 sin α sin β+cos2 α+cos2β+2 cos a cos β=a2+b2   1+1+2 (sin α sin β+cos α cos β)=a2+b2   2(sin α sin β+cos α cos β)=a2+b2

-2

   2cos (αβ)=a2+b22

Thus,  cos(αβ)=a2+b222      . . .(ii)

Again,

sin α+sin β=a  2 sin α+β2.cos αβ2=acos α+cos β=b   2 cos α+β2.cos αβ2=b
tanα+β2=ab      . . .(B)

Now,

sin(α+β)=2 tanα+β21+tan2(α+β2)=2ab1+a2b2=2aba2+b2

Thus,

sin(α+β)=2aba2+b2

(ii) cos (αβ)=a2+b222

We have,

sin α+sin β=a and cos α+cos β=b

Squaring and adding, we get

sin2α+sin2β+2sin α sin β+cos2 α+cos2β+2 cos α cos β=a2+b2  1+1+2 (sin α sin β+cos α cos β)=a2+b2  2(sin α sin β+cos α cos β)=a2+b2   2cos (αβ)=a2+b22Thus, cos(αβ)=a2+b222


Mathematics
RD Sharma
Standard XI

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