Prove that CotA/2-3cot3A/2=4sinA/1+2cosA?

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Answer 1 · 2018-02-11T19:42:57

Please refer to a Proof in the Explanation.

We know that, #tan3x=(3tanx-tan^3x)/(1-3tan^2x)#.

#:. cot3x=1/(tan3x)=(1-3tan^2x)/(3tanx-tan^3x)#.

Let, #tan(A/2)=t#. Then,

#cot(A/2)-3cot(3A/2)=1/t-(3(1-3t^2))/(3t-t^3)#,

#=1/t-(3(1-3t^2))/(t(3-t^2)#,

#={(3-t^2)-(3-9t^2)}/(t(3-t^2)#,

#=(8t)/(3-t^2)#,

#={(8t)/(1+t^2)}-:{(3-t^2)/(1+t^2)}#,

#={(8t)/(1+t^2)}-:{{(1+t^2)+(2-2t^2)}/(1+t^2)}#,

#={(8t)/(1+t^2)}-:{(1+t^2)/(1+t^2)+(2(1-t^2))/(1+t^2)}#,

#={4((2t)/(1+t^2))}-:{1+2((1-t^2)/(1+t^2))}#.

Since, #sinA=(2tan(A/2))/(1+tan^2(A/2)), and, cos2y=(1-tan^2(A/2))/(1+tan^2(A/2))#,

#cot(A/2)-3cot(3A/2)=4sinA-:(1+2cosA)#, as desired!