Verified answer

sin(x - 45) = sinx cos45 - cosx sin45 now

sin(x - 45) = √(2)sinx

sinx cos45 - cosx sin45 = √(2)sinx

sinx (1/√2) - cosx (1/√2) =√(2)sinx

1/√2 (sinx-cosx) = √2sinx

sinx-cosx=2sinx

1-cotx=2

cotx=-1

x=-pi/4 + n180 where n is iteger

Sin X 45

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2 sin x / cos x = 1 2 tanx = 1 tan x = 1/2 (1st and 3rd quadrants) x = 26.6° , 206.6° cos x.(sinx - cos x) = 0 cos x = 0, sin x = cos x cos x = 0 , tan x = 1 x = 90°, 270° , x = 45° , x = 225°

First, expand sin(x + 45)

sin(x + 45) =>

sin(x)cos(45) + sin(45)cos(x) =>

sin(x) * (sqrt(2)/2) + (sqrt(2)/2) * cos(x)

SO:

(sqrt(2)/2) * sin(x) + (sqrt(2)/2) * cos(x) = 2 * sin(x)

sqrt(2) * sin(x) + sqrt(2) * cos(x) = 4 * sin(x)

sin(x) + cos(x) = 2 * sqrt(2) * sin(x)

cos(x) = 2 * sqrt(2) * sin(x) - sin(x)

cos(x) = sin(x) * (2 * sqrt(2) - 1)

1 / (2 * sqrt(2) - 1) = sin(x) / cos(x)

1 / (2 * sqrt(2) - 1) = tan(x)