Question

Solve for x: cosx - sinx = 1

Answer 1

√2 (1/√2 cosx - 1/√2 sinx) = 1
sin(π/4 - x) = 1/√2
π/4 - x = π/4 or 3π/4
x = 0 or -π/2
in [0,2π] that would be 0 or 3π/2

Answer 2

First.

cos ( x + π / 4 )

Use formula:

cos ( a + b ) = cos a ∙ cos b - sin a ∙ sin b

cos ( x + π / 4 ) = cos x ∙ cos π / 4 - sinx ∙ sin π / 4

cos ( x + π / 4 ) = cos x ∙ 1 √2 - sinx ∙ 1 / √2

cos ( x + π / 4 ) = cos x / √2 - sinx / √2

cos x - sin x = 1

Dividie both sides by √2

cos x / √2 - sin x /√2 = 1 / √2

cos ( x + π / 4 ) = 1 / √2

Take inverse cosine of both sides.

Since the period of cos x is 2 π:

x + π / 4 = π / 4 ± 2 π n

Subtract π / 4 from both sides.

x = ± 2 π n

and

x + π / 4 = 2 π n + 7 π / 4

Subtract π / 4 from both sides.

x = 6 π / 4 ± 2 π n

x = 2 ∙ 3 ∙ π / 2 ∙ 2 ± 2 π n

x = 3 / 2 π ± 2 π n

where n = 0 , 1 , 2 , 3...