Question

If cos(A+2B)=m.cosA,show that

cotB=[(1+m)/(1-m)].tan(A+B)

Answer 1

Sol:

Given Cos A = m Cos B

⇒ Cos A / Cos B  = m / 1. ( apply componendo and dividendo)

⇒ (Cos A + Cos B) / (Cos A - Cos B)  = ( m + 1) / (m - 1)

⇒ {(2 Cos ( A+B) / 2 × Cos (A-B) / 2 } / {(- 2 Sin ( A+B) / 2 × Sin (A-B) / 2 }  = ( m + 1) / (m - 1)

⇒ {(2 Cos ( A+B) / 2 × Cos (B - A) / 2 } / {( 2 Sin ( A+B) / 2 × Sin (B - A) / 2 }  = ( m + 1) / (m - 1)

⇒ {( Cot ( A+B) / 2 × Cot (B - A) / 2 } = ( m + 1) / (m - 1)

⇒ {( Cot ( A+B) / 2 = { ( m + 1) / (m - 1) } Tan (B - A) / 2

Here x = ( m + 1) / (m - 1).