What is the general solution for csc^2(x)=(1/(cos(x)))^2 ?

The general solution for csc^2(x)=(1/(cos(x)))^2 is x=(3pi)/4+pin,x= pi/4+pin

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csc^2(x)=(1/(cos(x)))^2

csc^{2} (x) = (\frac{1}{cos ( x )})^{2}

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

Solution steps

csc^{2} (x) = (\frac{1}{cos ( x )})^{2}

Subtract

(\frac{1}{cos ( x )})^{2}

from both sides

csc^{2} (x) - \frac{1}{cos ^{2} ( x )} = 0

Simplify

csc^{2} (x) - \frac{1}{cos ^{2} ( x )} : \frac{csc ^{2} ( x ) cos ^{2} ( x ) - 1}{cos ^{2} ( x )}

\frac{csc ^{2} ( x ) cos ^{2} ( x ) - 1}{cos ^{2} ( x )} = 0

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

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

Factor

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

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

Solving each part separately

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

cos (x) csc (x) + 1 = 0 : x = \frac{3 π}{4} + πn

cos (x) csc (x) - 1 = 0 : x = \frac{π}{4} + πn

Combine all the solutions

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

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