Question

How do you solve the equation `1 + sin(x) = cos(x)? 1 + sin(x) = cos(x)`?

Answer 1

Use the identity: `color(blue)(sin x - cos x = (sqrt2)*sin (x - pi/4)`

`f(x) = 1 + sin x - cos x = 0`
`(sqrt2)*sin (x - pi/4) = -1`
`sin (x - pi/4) = -1/sqrt2`
There are 2 arcs that have the same sin value `(-1/(sqrt2))`

`x_1 - pi/4 = pi + pi/4 = (5pi)/4 -> x_1 = (5pi)/4 + pi/4 = (6pi)/4 = (3pi)/2`

`x_2 - pi/4 = 2pi - pi/4 = (7pi)/4 -> x_2 = (7pi)/4 + pi/4 = (8pi)/4 = 2pi.`

Check the 2 answers: `(3pi)/2` and `2pi`:

`x = (3pi)/2 -> sin x = -1`

and `cos x = 0 -> f(x) = 1 - 1 + 0 = 0.` Correct

`x = 2pi -> sin x = 0`,

and `cos x = 1 -> f(x) = 1 + 0 - 1 = 0.` Correct