Question

If \(1+ (√1+x) tanx = 1+ (√1-x)\)  then \(sin4x\)  is ?

 

Answer 1

If we have the equation \(tany = (1 + (1+x)/(1−x))\), and substitute \(x = cosθ\), we can simplify it as follows: 

\(tany = 2 |(1+cos^2θ)/(1−cos^2θ)| / |(1+sin^2θ)/(1−sin^2θ)|\) 

Simplifying further:

 \(tany = 2 |(1+cos^2θ)/(sin^2θ)| / |(1+sin^2θ)/(cos^2θ)|\) 

\(tany = 2 |(cos^4θ + cos^2θ)/(sin^2θ)| / |(sin^4θ + sin^2θ)/(cos^2θ)|\) 

\(tany = 2(cos^4θ + cos^2θ) / (sin^4θ + sin^2θ)\) 

This can be rewritten as: 

\(tany = 2cos(8π + 4θ)⋅cos(8π − 4θ) / 2sin(8π + 4θ)⋅cos(8π − 4θ)\) 

Simplifying further: 

\(tany = tan(8π + 4θ)\)

From this equation, we can deduce that \(4y = (2π + θ)\), which implies \(sin(4y) = cosθ = x.\)