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

The general solution for cos(x)=-sin^2(x) is x=2.23703…+2pin,x=-2.23703…+2pin

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

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

x = 2.23703 \dots + 2 πn, x = - 2.23703 \dots + 2 πn

Solution steps

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

Subtract

- sin^{2} (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

cos (x) = - \frac{- 1 + 5}{2}, cos (x) = \frac{1 + 5}{2}

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

cos (x) = \frac{1 + 5}{2} :

No Solution

Combine all the solutions

x = arccos (- \frac{- 1 + 5}{2}) + 2 πn, x = - arccos (- \frac{- 1 + 5}{2}) + 2 πn

Show solutions in decimal form

x = 2.23703 \dots + 2 πn, x = - 2.23703 \dots + 2 πn

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

x, 2 \cdot 2.25 \cdot cos (x) = 0

x, 2 sin (x) = 1

cot (x) + 2 cos (x) = 0

2 cot (θ) - 1 = 2, cot (θ) - 1 = 0

What is the general solution for cos(x)=-sin^2(x) ?
The general solution for cos(x)=-sin^2(x) is x=2.23703…+2pin,x=-2.23703…+2pin