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Question

If A+C=2B, then cosC-cosAsinA-sinC is equal to


A

cotB

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B

cot2B

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C

tan2B

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D

tanB

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Solution

The correct option is D

tanB


Explanation for the correct option:

Step 1. Find the value of cosC-cosAsinA-sinC:

Given,

A+C=2B

Now,

cosC-cosAsinA-sinC=-(cosAcosC)sinAsinC

=--2sin(A+C)2sin(AC)22cos(A+C)2sin(AC)2

Step 2. Use the identities:cosθ-cosϕ=-2sin(θ+ϕ)2sin(θ-ϕ)2,sinθ-sinϕ=2cos(θ+ϕ)2sin(θ-ϕ)2

cosC-cosAsinA-sinC=sin(A+C)2cos(A+C)2=sin2B2cos2B2=sinBcosB=tanB

cosC-cosAsinA-sinC=tanB

Hence, Option ‘D’ is Correct.


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