What is the general solution for 2tan^2(x)+1=cos(x) ?

The general solution for 2tan^2(x)+1=cos(x) is x=2pin

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2tan^2(x)+1=cos(x)

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

x = 2 πn

Solution steps

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

Square both sides

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

Subtract

cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Factor

(- 1 + 2 sec^{2} (x))^{2} - cos^{2} (x) : (- 1 + 2 sec^{2} (x) + cos (x)) (- 1 + 2 sec^{2} (x) - cos (x))

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

Solving each part separately

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

- 1 + 2 sec^{2} (x) + cos (x) = 0 : x = π + 2 πn

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

Combine all the solutions

x = π + 2 πn, x = 2 πn

Verify solutions by plugging them into the original equation

x = 2 πn

sec (x) = - \frac{4}{3}

sin (B) = \frac{1}{4}

55 sin (θ) - 20 - 20 cos (θ) = 0

y, sin (y) = 0

8 sin^{2} (x) = 1

What is the general solution for 2tan^2(x)+1=cos(x) ?
The general solution for 2tan^2(x)+1=cos(x) is x=2pin