What is the general solution for cos(2x)=5sin^2(x)-cos^2(x) ?

The general solution for cos(2x)=5sin^2(x)-cos^2(x) is x=(5pi)/6+pin,x= pi/6+pin

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cos(2x)=5sin^2(x)-cos^2(x)

cos (2 x) = 5 sin^{2} (x) - cos^{2} (x)

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Solution steps

cos (2 x) = 5 sin^{2} (x) - cos^{2} (x)

Subtract

5 sin^{2} (x) - cos^{2} (x)

from both sides

cos (2 x) - 5 sin^{2} (x) + cos^{2} (x) = 0

Rewrite using trig identities

2 cos^{2} (x) - 6 sin^{2} (x) = 0

Factor

2 cos^{2} (x) - 6 sin^{2} (x) : 2 (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

2 (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x)) = 0

Solving each part separately

cos (x) + 3 sin (x) = 0 or cos (x) - 3 sin (x) = 0

cos (x) + 3 sin (x) = 0 : x = \frac{5 π}{6} + πn

cos (x) - 3 sin (x) = 0 : x = \frac{π}{6} + πn

Combine all the solutions

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

3 sec (5 x) - 2 = 0

cot^{2} (x) sec (x) - sec (x) = 2 cot^{2} (x) - 2

5 cot^{2} (x) - 2 csc (x) + 2 = 0

\frac{sin ( 7 5 ^{\circ} )}{101} = \frac{sin ( b )}{65}

3 tan (A) + 5 = 0

What is the general solution for cos(2x)=5sin^2(x)-cos^2(x) ?
The general solution for cos(2x)=5sin^2(x)-cos^2(x) is x=(5pi)/6+pin,x= pi/6+pin