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

The general solution for cos(x)(cos(x)+5)=2sin^2(x) is x=1.23095…+2pin,x=2pi-1.23095…+2pin

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

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

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Solution steps

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

Subtract

2 sin^{2} (x)

from both sides

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

Rewrite using trig identities

- 2 + 3 cos^{2} (x) + 5 cos (x) = 0

Solve by substitution

cos (x) = \frac{1}{3}, cos (x) = - 2

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

cos (x) = - 2 :

No Solution

Combine all the solutions

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

tan^{2} (x) + 3 tan (x) - 10 = 0

cos^{2} (x) - sin (x) = - 1

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

cos^{2} (x) + 3 cos (x) = 1

cot^{2} (x) = \frac{tan ( x )}{2}

What is the general solution for cos(x)(cos(x)+5)=2sin^2(x) ?
The general solution for cos(x)(cos(x)+5)=2sin^2(x) is x=1.23095…+2pin,x=2pi-1.23095…+2pin