Question

If `sin θ + cos θ = sqrt(2) sin θ`, find cot θ.

Answer 1

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

Step-wise calculation:

1. `cos θ = sqrt(2) sin θ - sin θ`

= `(sqrt(2) - 1) sin θ`

2. Divide by sin θ (sin θ ≠ 0, since sin θ = 0 would force cos θ = 0 which is impossible):

`cot θ = cos θ/sin θ`

= `sqrt(2) - 1` 

Equivalently `tan θ = 1/(sqrt(2) - 1)`

= `sqrt(2) + 1` 

`cot θ = sqrt(2) - 1`