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Popular >

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

Solution

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

Solution

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

+1

Solution steps

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

Subtract

2 sin^{2} (\frac{x}{2})

from both sides

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

Rewrite using trig identities

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

Factor

- 2 sin^{2} (\frac{x}{2}) + 2 sin^{2} (x) : 2 (sin (x) + sin (\frac{x}{2})) (sin (x) - sin (\frac{x}{2}))

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

Solving each part separately

sin (x) + sin (\frac{x}{2}) = 0 or sin (x) - sin (\frac{x}{2}) = 0

sin (x) + sin (\frac{x}{2}) = 0 : x = 2 π + 8 πn, x = 6 π + 8 πn, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (x) - sin (\frac{x}{2}) = 0 : x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

Combine all the solutions

x = 2 π + 8 πn, x = 6 π + 8 πn, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}, x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

Merge Overlapping Intervals

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

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