Question

if tan2Atan4A=1 then prove sin2A=1/2​

Answer 1

Given---> tan2A tan4A = 1

To prove ---> Sin2A = 1 / 2

Proof---> ATQ,

tan2A tan4A = 1

=> tan4A = 1 / tan2A

We know that, Cotθ = 1 / tanθ , applying it here we get,

=> tan4A = Cot2A

We know that, tan ( 90° - θ ) = Cotθ , applying it here , we get,

=> tan4A = tan ( 90° - 2A )

=> 4A = 90° - 2A

=> 4A + 2A = 90°

=> 6A = 90°

=> A = 90° / 6

=> A = 15°

LHS = Sin2A

= Sin2 ( 15° )

= Sin 30°

= 1 / 2 = RHS

Additional information--->

1) Sin²θ + Cos²θ = 1

2) 1 + tan²θ = Sec²θ

3) 1 + Cot²θ = Cosec²θ