Solution to a trigonometric problem requested by one student

Prob: if tan (x)= a/b then find      { a sin (2X) + b cos (2X) }

Sol: when tan (x) = a/b ( opp side is 'a' units and adj side is 'b' units ) then sin x = a/√a²+b² and cos x= b/√a²+b² ( because hyp will be √a²+b² )

a sin (2X) = a ( 2 sin x cos x ) = 2 a { (a/√a²+b²) (b/√a²+b²) } = 2 a { a b/a²+b² } 

= 2 a²b/(a²+b²) -----------------------(A)

b cos (2X)= b ( 2 cos² x - 1) = b { 2 b²/(a² + b²) - 1 } = { 2 b³ / (a² + b² ) - b }

= {2 b³ - ( a²b +  b³ ) } /(a² + b² ) = (2 b³ - a²b - b³ )/(a² + b ²)---------------------(B)

Adding (A) + (B)

= { 2 a²b/( a² + b² ) }+{ (2 b³ - a²b - b³)/( a² + b² ) } 

=( 2 a²b + 2 b³ - a²b - b³)/( a² + b² )

=( 2 a²b - a²b + 2 b³ - b³ )/(a² + b² )

= ( a²b + b³ )/( a² + b² )

= b ( a² + b² ) /( a² + b² )

= b 

Hence a sin (2X) + b cos (2x) = b  when tan (x) = a/b