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 $sin(4y) = cosθ = x.$