
To establish:the given identity.

Answer to Problem 70RE
The identity is established.
Explanation of Solution
Given:
The given identity is sin(2θ)+sin(4θ)sin(2θ)−sin(4θ)+tan(3θ)tanθ=0 .
Calculation:
As the given identity is sin(2θ)+sin(4θ)sin(2θ)−sin(4θ)+tan(3θ)tanθ=0 .
Now, the identity is established when sin(2θ)+sin(4θ)sin(2θ)−sin(4θ)=−tan(3θ)tanθ :
tan(3θ)tanθ=sin(3θ)cos(3θ)sinθcosθ =sin(3θ)cosθcos(3θ)sinθ =12[sin(3θ+θ)+sin(3θ−θ)]12[sin(θ+3θ)+sin(θ−3θ)] =[sin(4θ)+sin(2θ)][sin(4θ)+sin(−2θ)] =sin(4θ)+sin(2θ)sin(4θ)−sin(2θ) =−[sin(4θ)+sin(2θ)sin(2θ)−sin(4θ)] ......(1)
Now, put the value of (1) in the identity, then
sin(2θ)+sin(4θ)sin(2θ)−sin(4θ)+[−sin(2θ)+sin(4θ)sin(2θ)−sin(4θ)]sin(2θ)+sin(4θ)sin(2θ)−sin(4θ)−sin(2θ)+sin(4θ)sin(2θ)−sin(4θ)=0
Thus, sin(2θ)+sin(4θ)sin(2θ)−sin(4θ)+tan(3θ)tanθ=0
Hence, the identity is established.
Chapter 7 Solutions
Precalculus
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