Question

If `(cos θ - sin θ) = sqrt(2) sin θ` then prove that `(cos θ + sin θ) = sqrt(2) cos θ`.

Answer 1

Given: `cos θ - sin θ = sqrt(2) sin θ`

To Prove: `cos θ + sin θ = sqrt(2) cos θ`

Proof [Step-wise]:

1. From the given,

`cos θ - sin θ = sqrt(2) sin θ` 

⇒ `cos θ = (sqrt(2) + 1) sin θ`

2. Check division is valid:

If cos θ = 0 then the given becomes `-sin θ = sqrt(2) sin θ` 

⇒ `(sqrt(2) + 1) sin θ = 0`

⇒ sin θ = 0, which is impossible.

Hence cos θ ≠ 0.

3. Divide `cos θ = (sqrt(2) + 1) sin θ` by cos θ to get `1 = (sqrt(2) + 1) tan θ`, so `tan θ = 1/(sqrt(2) + 1)`.

4. Rationalize `1/(sqrt(2) + 1)`:

`1/(sqrt(2) + 1) = sqrt(2) - 1`

Thus `tan θ = sqrt(2) - 1`.

5. Compute `(cos θ + sin θ)/cos θ = 1 + tan θ`

= `1 + (sqrt(2) - 1)`

= `sqrt(2)`

6. Multiply both sides by cos θ:

`cos θ + sin θ = sqrt(2) cos θ`, as required.

Therefore, from `cos θ - sin θ = sqrt(2) sin θ` we have proven `cos θ + sin θ = sqrt(2) cos θ`.