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Question

If cos α+cos β=13 and sin α+sin β=14,  prove that cosαβ2=±524


Solution

We have,

cos α+cos β=13 and sin α+sin β=14

Squaring and adding, we get

(cos2 α+cos2β+2cos α cos β)+(sin2α+sin2β+2sin α sin β)=19+116    1+1+2(cos α+cos β+sin α sin β)=25144    2 cos (αβ)=251442=263144    cos (αβ)=263288

Now,

cos (αβ2)=1+cos(αβ)2=12632882=25576=±524     cos(αβ2)=±524


Mathematics
RD Sharma
Standard XI

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