What is the general solution for cos^2(x)-sin^2(x)=sec^2(x)-csc^2(x) ?

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

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cos^2(x)-sin^2(x)=sec^2(x)-csc^2(x)

cos^{2} (x) - sin^{2} (x) = sec^{2} (x) - csc^{2} (x)

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

Solution steps

cos^{2} (x) - sin^{2} (x) = sec^{2} (x) - csc^{2} (x)

Subtract

sec^{2} (x) - csc^{2} (x)

from both sides

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

Express with sin, cos

\frac{cos ^{2} ( x ) - sin ^{2} ( x ) - cos ^{2} ( x ) sin ^{4} ( x ) + cos ^{4} ( 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^{4} (x) + cos^{4} (x) sin^{2} (x) = 0

Factor

cos^{2} (x) - sin^{2} (x) - cos^{2} (x) sin^{4} (x) + cos^{4} (x) sin^{2} (x) : (1 + cos^{2} (x) sin^{2} (x)) (cos (x) + sin (x)) (cos (x) - sin (x))

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

Solving each part separately

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

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

No Solution

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

cos (x) - sin (x) = 0 : x = \frac{π}{4} + πn

Combine all the solutions

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

2 - 3 sin (x) = 0

y = cos (x), x

\frac{cos ( θ )}{sin ( θ )} = 0

sin (2 x) = sin (x), 0 \leq x < 2 π

tan (θ) = \frac{5}{5 3}

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