Prove Basic Problems (Dealing with Sin, Cos, Tan)

Question

Prove:

 

SQUARE ROOT OF (3cosx+4sinx)²+(4cosx-3sinx)² = 5

 

Please include step by step and the identities used next to each step! :)

Answer 1

Left side = √(9cos²x+24cosxsinx+16sin²x+16cos²x-24cosxsinx+9sin²x)

combine like terms:

=√(25cos²x+25sin²x) 

= √25 (Since sin²x+cos²x=1)

=5

=right side