What is the general solution for sin^2(a)+1/(sec(a))= 5/4 ?

The general solution for sin^2(a)+1/(sec(a))= 5/4 is a= pi/3+2pin,a=(5pi)/3+2pin

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sin^2(a)+1/(sec(a))= 5/4

sin^{2} (a) + \frac{1}{sec ( a )} = \frac{5}{4}

a = \frac{π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn

Solution steps

sin^{2} (a) + \frac{1}{sec ( a )} = \frac{5}{4}

Subtract

\frac{5}{4}

from both sides

sin^{2} (a) + \frac{1}{sec ( a )} - \frac{5}{4} = 0

Simplify

sin^{2} (a) + \frac{1}{sec ( a )} - \frac{5}{4} : \frac{4 sin ^{2} ( a ) sec ( a ) + 4 - 5 sec ( a )}{4 sec ( a )}

\frac{4 sin ^{2} ( a ) sec ( a ) + 4 - 5 sec ( a )}{4 sec ( a )} = 0

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

4 sin^{2} (a) sec (a) + 4 - 5 sec (a) = 0

Express with sin, cos

4 sin^{2} (a) \frac{1}{cos ( a )} + 4 - 5 \cdot \frac{1}{cos ( a )} = 0

Simplify

4 sin^{2} (a) \frac{1}{cos ( a )} + 4 - 5 \cdot \frac{1}{cos ( a )} : \frac{4 sin ^{2} ( a ) - 5 + 4 cos ( a )}{cos ( a )}

\frac{4 sin ^{2} ( a ) - 5 + 4 cos ( a )}{cos ( a )} = 0

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

4 sin^{2} (a) - 5 + 4 cos (a) = 0

Subtract

4 cos (a)

from both sides

4 sin^{2} (a) - 5 = - 4 cos (a)

Square both sides

(4 sin^{2} (a) - 5)^{2} = (- 4 cos (a))^{2}

Subtract

(- 4 cos (a))^{2}

from both sides

(4 sin^{2} (a) - 5)^{2} - 16 cos^{2} (a) = 0

Factor

(4 sin^{2} (a) - 5)^{2} - 16 cos^{2} (a) : (4 sin^{2} (a) - 5 + 4 cos (a)) (4 sin^{2} (a) - 5 - 4 cos (a))

(4 sin^{2} (a) - 5 + 4 cos (a)) (4 sin^{2} (a) - 5 - 4 cos (a)) = 0

Solving each part separately

4 sin^{2} (a) - 5 + 4 cos (a) = 0 or 4 sin^{2} (a) - 5 - 4 cos (a) = 0

4 sin^{2} (a) - 5 + 4 cos (a) = 0 : a = \frac{π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn

4 sin^{2} (a) - 5 - 4 cos (a) = 0 : a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn

Combine all the solutions

a = \frac{π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn, a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn

Verify solutions by plugging them into the original equation

a = \frac{π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn

\frac{csc ( x )}{cot ( x )} = 2

sin (4 x) + 6 cos (2 x) = 0

sec (θ) + tan (θ) = 0

sin (θ) = \frac{3.5}{5}

sec (a) = \frac{41}{12}, 2 π < a < \frac{5 π}{2}, sin (\frac{a}{2})

What is the general solution for sin^2(a)+1/(sec(a))= 5/4 ?
The general solution for sin^2(a)+1/(sec(a))= 5/4 is a= pi/3+2pin,a=(5pi)/3+2pin