What is the general solution for csc(x)cot(x)=sqrt(2) ?

The general solution for csc(x)cot(x)=sqrt(2) is x= pi/4+2pin,x=(7pi)/4+2pin

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csc(x)cot(x)=sqrt(2)

csc (x) cot (x) = 2

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Solution steps

csc (x) cot (x) = 2

Subtract

2

from both sides

csc (x) cot (x) - 2 = 0

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Solve by substitution

cos (x) = \frac{2}{2}, cos (x) = - 2

cos (x) = \frac{2}{2} : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

cos (x) = - 2 :

No Solution

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

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

2 cos (x) = 2 sin (x)

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

tan (θ) = \frac{7}{4}

What is the general solution for csc(x)cot(x)=sqrt(2) ?
The general solution for csc(x)cot(x)=sqrt(2) is x= pi/4+2pin,x=(7pi)/4+2pin