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Question

In Δ, prove that cos2A2+cos2B2cos2C2=2cosA2cosB2sinC2

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Solution

Given, cos2A2+cos2B2cos2C2=2cosA2.cosB2.cosC2
Taking L.H.S
A+B+C=π
=cos2A2+cos2B2cos2C2
=1sin2A2+cos2B2cos2C2
=1cos2B2sin2A2cos2C2
=1+cosB+A2.cosAB2cos2C2
=1+sinC2.cosAB21+sin2C2
=sinC2[cosAB2+sin(π2(A+B2)]
=sinC2[cosAB2+cos(A+B2]
=sinC2[2cosA2.cosB2]
=2cosA2.cosB2.sinC2

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