What is the general solution for cos(θ)=tan(θ) ?

The general solution for cos(θ)=tan(θ) is θ=0.66623…+2pin,θ=pi-0.66623…+2pin

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cos(θ)=tan(θ)

cos (θ) = tan (θ)

θ = 0.66623 \dots + 2 πn, θ = π - 0.66623 \dots + 2 πn

Solution steps

cos (θ) = tan (θ)

Subtract

tan (θ)

from both sides

cos (θ) - tan (θ) = 0

Express with sin, cos

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos^{2} (θ) - sin (θ) = 0

Rewrite using trig identities

1 - sin (θ) - sin^{2} (θ) = 0

Solve by substitution

sin (θ) = - \frac{1 + 5}{2}, sin (θ) = \frac{5 - 1}{2}

sin (θ) = - \frac{1 + 5}{2} :

No Solution

sin (θ) = \frac{5 - 1}{2} : θ = arcsin (\frac{5 - 1}{2}) + 2 πn, θ = π - arcsin (\frac{5 - 1}{2}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{5 - 1}{2}) + 2 πn, θ = π - arcsin (\frac{5 - 1}{2}) + 2 πn

Show solutions in decimal form

θ = 0.66623 \dots + 2 πn, θ = π - 0.66623 \dots + 2 πn

tan (\frac{π}{4} + x) = \frac{1 - tan ( x )}{1 + tan ( x )}

cos (x) = \frac{45}{52}

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

6 tan^{2} (x) + 8 = 10

2 sin (x - \frac{π}{4}) + 2 cos (x) = 1

What is the general solution for cos(θ)=tan(θ) ?
The general solution for cos(θ)=tan(θ) is θ=0.66623…+2pin,θ=pi-0.66623…+2pin