Question

If 3sinθ+5cosθ=5, prove that 5 sinθ-3cosθ=±3.

Answer 1

3 sinθ + 5 cosθ = 5

Squaring both sides ,
( 3 sinθ + 5 cosθ )² = 5²

9 sin²θ + 25 cos²θ + 30 sinθ.cos θ = 25   

9( 1 - cos²θ ) + 25( 1 -sin²θ ) + 30 sinθ.cosθ  =25

Using identity , [ sin²θ + cos²θ =1 ]

9 - 9 cos²θ + 25 - 25 sin²θ +30 sinθ.cosθ =25

- 9 cos²θ  - 25 sin²θ + 30 sinθcosθ = - 9

25 sin²θ + 9 cos²θ - 30 sinθ.cosθ = 9

( 5.sinθ - 3.cosθ )² = 9

[ 5 sinθ - 3 cosθ ) = ±3 ]

Answer 2

Refer to the attached picture