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

The general solution for sec^2(θ)(1+cos^2(θ))=2-tan^2(θ) is θ=pin

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sec^2(θ)(1+cos^2(θ))=2-tan^2(θ)

sec^{2} (θ) (1 + cos^{2} (θ)) = 2 - tan^{2} (θ)

θ = πn

Solution steps

sec^{2} (θ) (1 + cos^{2} (θ)) = 2 - tan^{2} (θ)

Subtract

2 - tan^{2} (θ)

from both sides

sec^{2} (θ) + sec^{2} (θ) cos^{2} (θ) - 2 + tan^{2} (θ) = 0

Express with sin, cos

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

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

1 - cos^{2} (θ) + sin^{2} (θ) = 0

Rewrite using trig identities

1 - cos (2 θ) = 0

Move

1

to the right side

- cos (2 θ) = - 1

Divide both sides by

- 1

cos (2 θ) = 1

General solutions for

cos (2 θ) = 1

2 θ = 0 + 2 πn

Solve

2 θ = 0 + 2 πn : θ = πn

θ = πn

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

8 tan (x) - 12 = 5 tan (x) - 9

64 = arctan (\frac{x}{11})

2 sin (3 x) cos (3 x) = 1

sin (x) = \frac{3 10}{10}

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