What is the general solution for sqrt(2)=sec(x)tan(x) ?

The general solution for sqrt(2)=sec(x)tan(x) is x= pi/4+2pin,x=(3pi)/4+2pin

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sqrt(2)=sec(x)tan(x)

2 = sec (x) tan (x)

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Solution steps

2 = sec (x) tan (x)

Switch sides

sec (x) tan (x) = 2

Subtract

2

from both sides

sec (x) tan (x) - 2 = 0

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Solve by substitution

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

sin (x) = \frac{2}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - 2 :

No Solution

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

cos (x) = \frac{9}{13}

cos^{2} (x) = \frac{( 6 k - 9 )}{10}

cot (x) + csc (x) = - 1

cos^{2} (x) = \frac{( 3 k - 3 )}{8}

8 sin^{2} (x) - 10 cos (x) - 11 = 0

What is the general solution for sqrt(2)=sec(x)tan(x) ?
The general solution for sqrt(2)=sec(x)tan(x) is x= pi/4+2pin,x=(3pi)/4+2pin