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George Bray

George Bray

Answered question

2022-06-21

Given [ 1 + ( 1 + x ) 1 / 2 ] × tan ( x ) = [ 1 + ( 1 x ) 1 / 2 ] , find sin 4 x

Answer & Explanation

hildiadau0o

hildiadau0o

Beginner2022-06-22Added 21 answers

Let:
t = tan ( x ) = 1 + 1 x 1 + 1 + x
Using the double-angle sine and half-angle tangent formulas:
sin ( 4 x ) = 2 sin ( 2 x ) cos ( 2 x ) = 2 2 t 1 + t 2 1 t 2 1 + t 2 = 4 t ( 1 t 2 ) ( 1 + t 2 ) 2
Substituting back t in terms of x and simplifying:
sin ( 4 x ) = 2 x ( 1 x 2 + 2 1 x + 2 1 + x + 3 ) ( 1 x + 1 + x + 2 ) 2 = 2 x ( 1 x 2 + 2 1 x + 2 1 + x + 3 ) ( 1 x ) + ( 1 + x ) + 4 + 2 ( 1 x ) ( 1 + x ) + 4 1 x + 4 1 + x = 2 x ( 1 x 2 + 2 1 x + 2 1 + x + 3 ) 2 ( 3 + 1 x 2 + 1 x + 1 + x ) = x

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