What is the general solution for tan(θ)+sec(θ)=2cos(θ) ?

The general solution for tan(θ)+sec(θ)=2cos(θ) is θ= pi/6+2pin,θ=(5pi)/6+2pin

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tan(θ)+sec(θ)=2cos(θ)

tan (θ) + sec (θ) = 2 cos (θ)

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

Solution steps

tan (θ) + sec (θ) = 2 cos (θ)

Subtract

2 cos (θ)

from both sides

tan (θ) + sec (θ) - 2 cos (θ) = 0

Express with sin, cos

\frac{1 + sin ( θ ) - 2 cos ^{2} ( θ )}{cos ( θ )} = 0

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

1 + sin (θ) - 2 cos^{2} (θ) = 0

Rewrite using trig identities

- 1 + sin (θ) + 2 sin^{2} (θ) = 0

Solve by substitution

sin (θ) = \frac{1}{2}, sin (θ) = - 1

sin (θ) = \frac{1}{2} : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

sin (θ) = - 1 : θ = \frac{3 π}{2} + 2 πn

Combine all the solutions

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = \frac{3 π}{2} + 2 πn

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

cos (x) = 0.68

cos (x) = 0.65

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

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

tan (θ) = 30.48 (80)

What is the general solution for tan(θ)+sec(θ)=2cos(θ) ?
The general solution for tan(θ)+sec(θ)=2cos(θ) is θ= pi/6+2pin,θ=(5pi)/6+2pin