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# cos A=m cos B, then prove that cot (A+B/2)=( m+1/m-1)tan(B-A/2)

Grade:10

Vikas TU
14149 Points
4 years ago

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).

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