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

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

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sec(x)csc(x)=sqrt(2)sec(x)

sec (x) csc (x) = 2 sec (x)

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

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

sec (x) csc (x) = 2 sec (x)

Subtract

2 sec (x)

from both sides

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

Factor

sec (x) csc (x) - 2 sec (x) : sec (x) (csc (x) - 2)

sec (x) csc (x) - 2 sec (x)

Factor out common term

sec (x)

= sec (x) (csc (x) - 2)

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

Solving each part separately

sec (x) = 0 or csc (x) - 2 = 0

sec (x) = 0 :

No Solution

sec (x) = 0

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

csc (x) - 2 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) - 2 = 0

Move

2

to the right side

csc (x) - 2 = 0

Add

2

to both sides

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

Simplify

csc (x) = 2

csc (x) = 2

General solutions for

csc (x) = 2

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

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

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

Combine all the solutions

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

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

(tan (θ) - 1) (cos (θ) - 1) = 0

cos (x) = \frac{16}{20}

cos (2 x) + 10 sin^{2} (x) = 7

cos (x) = - 3

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