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

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

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sec^2(x)+csc^2(x)=1

sec^{2} (x) + csc^{2} (x) = 1

No Solution for x \in R

Solution steps

sec^{2} (x) + csc^{2} (x) = 1

Subtract

1

from both sides

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

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

1 + sin (x) cos (x) = 0 :

No Solution

1 - sin (x) cos (x) = 0 :

No Solution

Combine all the solutions

No Solution for x \in R

sin (x) = 0.653

- 50 cos (x) - 86.6025400000000 \dots sin (x) = - 50

sin (\frac{1}{2} x) + 1 = 0

4 cos^{2} (x) + 8 cos (x) + 4 = 0

4 sin (x) cos (x) = 22 cos (x)

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