What is the general solution for cot^2(x)=sec^2(x)-1 ?

The general solution for cot^2(x)=sec^2(x)-1 is x= pi/4+pin,x=(3pi)/4+pin

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cot^2(x)=sec^2(x)-1

cot^{2} (x) = sec^{2} (x) - 1

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Solution steps

cot^{2} (x) = sec^{2} (x) - 1

Subtract

sec^{2} (x) - 1

from both sides

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

Rewrite using trig identities

cot^{2} (x) - tan^{2} (x) = 0

Factor

cot^{2} (x) - tan^{2} (x) : (cot (x) + tan (x)) (cot (x) - tan (x))

(cot (x) + tan (x)) (cot (x) - tan (x)) = 0

Solving each part separately

cot (x) + tan (x) = 0 or cot (x) - tan (x) = 0

cot (x) + tan (x) = 0 :

No Solution

cot (x) - tan (x) = 0 : x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

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

7 sin^{2} (x) + 2 sin^{2} (x) - 3 cos^{2} (x) = 0

sin^{2} (x) - cos^{2} (x) = cos^{4} (x)

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

\frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} = 1

What is the general solution for cot^2(x)=sec^2(x)-1 ?
The general solution for cot^2(x)=sec^2(x)-1 is x= pi/4+pin,x=(3pi)/4+pin