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

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

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

2 = csc (x) cot (x)

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

Solution steps

2 = csc (x) cot (x)

Switch sides

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

sin^{2} (x) = 5

sin^{2} (x) = a

θ, sin (2 θ) = cos (θ)

5 cos^{2} (θ) + 32 sin (θ) - \frac{11}{2} = 0

cos^{2} (x) - 5 cos (x) + 2 = 0

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