What is the general solution for (9sin(2x)+9cos(2x))^2=81 ?

The general solution for (9sin(2x)+9cos(2x))^2=81 is x=pin+pi/2 ,x=pin+(3pi)/4 ,x=pin,x=pin+pi/4

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(9sin(2x)+9cos(2x))^2=81

(9 sin (2 x) + 9 cos (2 x))^{2} = 81

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

Solution steps

(9 sin (2 x) + 9 cos (2 x))^{2} = 81

Subtract

81

from both sides

(9 sin (2 x) + 9 cos (2 x))^{2} - 81 = 0

Factor

(9 sin (2 x) + 9 cos (2 x))^{2} - 81 : 81 (sin (2 x) + cos (2 x) + 1) (sin (2 x) + cos (2 x) - 1)

81 (sin (2 x) + cos (2 x) + 1) (sin (2 x) + cos (2 x) - 1) = 0

Solving each part separately

sin (2 x) + cos (2 x) + 1 = 0 or sin (2 x) + cos (2 x) - 1 = 0

sin (2 x) + cos (2 x) + 1 = 0 : x = πn + \frac{π}{2}, x = πn + \frac{3 π}{4}

sin (2 x) + cos (2 x) - 1 = 0 : x = πn, x = πn + \frac{π}{4}

Combine all the solutions

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

- 2 cos^{2} (θ) + sin (θ) - 3 = - 4

sec^{2} (x) - 9 sec (x) = 0

- 3 sin (2 θ) = 5 cos (2 θ)

(tan^{2} (θ) - 4) (2 cos (θ) + 1) = 0

cot (t) = - 1

What is the general solution for (9sin(2x)+9cos(2x))^2=81 ?
The general solution for (9sin(2x)+9cos(2x))^2=81 is x=pin+pi/2 ,x=pin+(3pi)/4 ,x=pin,x=pin+pi/4