What is the general solution for sqrt(cos^2(2x)+1)+sin(2x)=2 ?

The general solution for sqrt(cos^2(2x)+1)+sin(2x)=2 is x= pi/4+pin

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sqrt(cos^2(2x)+1)+sin(2x)=2

cos^{2} (2 x) + 1 + sin (2 x) = 2

x = \frac{π}{4} + πn

Solution steps

cos^{2} (2 x) + 1 + sin (2 x) = 2

Subtract

2

from both sides

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

Rewrite using trig identities

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

Solve by substitution

sin (2 x) = 1

sin (2 x) = 1 : x = \frac{π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn

8 sin^{2} (x) 2 cos (x) = 7, 0 \leq x \leq 2 π

2 sin^{2} (x) - sin (x) + 3 = 4

9 cos (2 x) = 9 cos^{2} (x) - 4

2 - cos (x) = 0

sin (\frac{1}{x}) = 1

What is the general solution for sqrt(cos^2(2x)+1)+sin(2x)=2 ?
The general solution for sqrt(cos^2(2x)+1)+sin(2x)=2 is x= pi/4+pin