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

The general solution for sec^2(θ)+csc^2(θ)=1 is No Solution for θ\in\mathbb{R}

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sec^2(θ)+csc^2(θ)=1

sec^{2} (θ) + csc^{2} (θ) = 1

No Solution for θ \in R

Solution steps

sec^{2} (θ) + csc^{2} (θ) = 1

Subtract

1

from both sides

sec^{2} (θ) + csc^{2} (θ) - 1 = 0

Express with sin, cos

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

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

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

Rewrite using trig identities

1 - cos^{2} (θ) sin^{2} (θ) = 0

Factor

1 - cos^{2} (θ) sin^{2} (θ) : (1 + sin (θ) cos (θ)) (1 - sin (θ) cos (θ))

(1 + sin (θ) cos (θ)) (1 - sin (θ) cos (θ)) = 0

Solving each part separately

1 + sin (θ) cos (θ) = 0 or 1 - sin (θ) cos (θ) = 0

1 + sin (θ) cos (θ) = 0 :

No Solution

1 - sin (θ) cos (θ) = 0 :

No Solution

Combine all the solutions

No Solution for θ \in R

cos (t) - sin (t) = 0

sin (x) - cos (x) = \frac{1}{3}

tan^{2} (x) = sec (x) - 1, 0 \leq x \leq 2 π

sin^{2} (θ) = 2 sin^{2} (\frac{θ}{2})

cos (x) = cos (x)

What is the general solution for sec^2(θ)+csc^2(θ)=1 ?
The general solution for sec^2(θ)+csc^2(θ)=1 is No Solution for θ\in\mathbb{R}