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Popular >

2-2cos^2(x/2)=2sin^2(x)

Solution

2 - 2 cos^{2} (\frac{x}{2}) = 2 sin^{2} (x)

Solution

x = 4 πn, x = 2 π + 4 πn, x = \frac{4 π}{3} + 4 πn, x = \frac{8 π}{3} + 4 πn, x = \frac{2 π + 8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

+1

Solution steps

2 - 2 cos^{2} (\frac{x}{2}) = 2 sin^{2} (x)

Subtract

2 sin^{2} (x)

from both sides

2 - 2 cos^{2} (\frac{x}{2}) - 2 sin^{2} (x) = 0

Let:

u = \frac{x}{2}

2 - 2 cos^{2} (u) - 2 sin^{2} (2 u) = 0

Rewrite using trig identities

- 2 sin^{2} (2 u) + 2 sin^{2} (u) = 0

Factor

- 2 sin^{2} (2 u) + 2 sin^{2} (u) : 2 (sin (u) + sin (2 u)) (sin (u) - sin (2 u))

2 (sin (u) + sin (2 u)) (sin (u) - sin (2 u)) = 0

Solving each part separately

sin (u) + sin (2 u) = 0 or sin (u) - sin (2 u) = 0

sin (u) + sin (2 u) = 0 : u = 2 πn, u = π + 2 πn, u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

sin (u) - sin (2 u) = 0 : u = \frac{π}{3} + \frac{4 πn}{3}, u = π + \frac{4 πn}{3}, u = 4 πn, u = 2 π + 4 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn, u = \frac{π}{3} + \frac{4 πn}{3}, u = π + \frac{4 πn}{3}, u = 4 πn, u = 2 π + 4 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

\frac{x}{2} = \frac{2 π}{3} + 2 πn : x = \frac{4 π}{3} + 4 πn

\frac{x}{2} = \frac{4 π}{3} + 2 πn : x = \frac{8 π}{3} + 4 πn

\frac{x}{2} = \frac{π}{3} + \frac{4 πn}{3} : x = \frac{2 π + 8 πn}{3}

\frac{x}{2} = π + \frac{4 πn}{3} : x = 2 π + \frac{8 πn}{3}

\frac{x}{2} = 4 πn : x = 8 πn

\frac{x}{2} = 2 π + 4 πn : x = 4 π + 8 πn

x = 4 πn, x = 2 π + 4 πn, x = \frac{4 π}{3} + 4 πn, x = \frac{8 π}{3} + 4 πn, x = \frac{2 π + 8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

Merge Overlapping Intervals

x = 4 πn, x = 2 π + 4 πn, x = \frac{4 π}{3} + 4 πn, x = \frac{8 π}{3} + 4 πn, x = \frac{2 π + 8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

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