What is the general solution for 2+3cos^2(x)=sin^2(x)+4cos(x) ?

The general solution for 2+3cos^2(x)=sin^2(x)+4cos(x) is x= pi/3+2pin,x=(5pi)/3+2pin

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2+3cos^2(x)=sin^2(x)+4cos(x)

2 + 3 cos^{2} (x) = sin^{2} (x) + 4 cos (x)

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

2 + 3 cos^{2} (x) = sin^{2} (x) + 4 cos (x)

Subtract

sin^{2} (x) + 4 cos (x)

from both sides

2 + 3 cos^{2} (x) - sin^{2} (x) - 4 cos (x) = 0

Rewrite using trig identities

1 - 4 cos (x) + 4 cos^{2} (x) = 0

Solve by substitution

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

tan (4 x) - tan (2 x) = 0

t, sin (t) + 2 sin (2 t) = 0

tan^{2} (x) - sin (x) tan^{2} (x) = 0

- 1 - tan (x) = sec (x), 0 \leq x < 2 π

What is the general solution for 2+3cos^2(x)=sin^2(x)+4cos(x) ?
The general solution for 2+3cos^2(x)=sin^2(x)+4cos(x) is x= pi/3+2pin,x=(5pi)/3+2pin