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

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

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cos(x)=tan(x)

cos (x) = tan (x)

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

Solution steps

cos (x) = tan (x)

Subtract

tan (x)

from both sides

cos (x) - tan (x) = 0

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Solve by substitution

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

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

No Solution

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

Combine all the solutions

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

Show solutions in decimal form

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

2 sin (x) = sin (x)

- sin (2 θ) - cos (4 θ) = 0

5 tan (θ) + 9 = 0

3 sin (θ) = 3 sin (θ) + 3 cos (θ) + 3 sin (θ)

2 sin^{2} (x) - 3 sin (x) = 2

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