Question

If sin thita + 2cos thita = 1 prove that 2sin thita - cos thita =2

Answer 1

sin∅ + 2cos∅ = 1
squaring both sides ,
sin²∅ + 4cos²∅ + 4sin∅.cos∅ = 1 ------(1)

Let
2sin∅ - cos∅ ≠ 2
Let, 2sin∅ - cos∅ = P
now,
squaring both sides ,
4sin²∅ + cos²∅ - 4sin∅.cos∅ = P² ------(2)

adding equations (1) and (2)

sin²∅ + 4cos²∅ + 4sin∅.cos∅ + 4sin²∅ + cos²∅ - 4sin∅.cos∅ = P² + 1
(sin²∅ + cos²∅) + 4(sin²∅ + cos²∅) = P² + 1

we know, by trigonometric identities
sin²x + cos²x = 1 use this here,

1 + 4 × 1 = P² + 1
P² = 4
hence, P = ±2
hence, 2sin∅ - cos∅ = ±2

so, our assumption is wrong that
2sin∅ - cos∅ ≠ 2 ,

hence, 2sin∅ - cos∅ = 2