What is the general solution for 2sin^2(45-a)=1-sin^2(a) ?

The general solution for 2sin^2(45-a)=1-sin^2(a) is a=360n,a=180+360n,a=1.10714…+180n

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2sin^2(45-a)=1-sin^2(a)

2 sin^{2} (4 5^{\circ} - a) = 1 - sin^{2} (a)

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 1.10714 \dots + 18 0^{\circ} n

Solution steps

2 sin^{2} (4 5^{\circ} - a) = 1 - sin^{2} (a)

Rewrite using trig identities

(cos (a) - sin (a))^{2} = 1 - sin^{2} (a)

Subtract

1 - sin^{2} (a)

from both sides

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

Rewrite using trig identities

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

Factor

sin^{2} (a) - 2 cos (a) sin (a) : sin (a) (sin (a) - 2 cos (a))

sin (a) (sin (a) - 2 cos (a)) = 0

Solving each part separately

sin (a) = 0 or sin (a) - 2 cos (a) = 0

sin (a) = 0 : a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

sin (a) - 2 cos (a) = 0 : a = arctan (2) + 18 0^{\circ} n

Combine all the solutions

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = arctan (2) + 18 0^{\circ} n

Show solutions in decimal form

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 1.10714 \dots + 18 0^{\circ} n

(2 cos (a) + 1) (2 cos (a) - 1) = 2 cos^{2} (a) + 1

tan^{2} (x) cot (x) = 1

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

6 cos^{2} (x) - 5 sin (x) + 5 = 0

1 + cos^{2} (a) = 3 sin (x) cos (x)

What is the general solution for 2sin^2(45-a)=1-sin^2(a) ?
The general solution for 2sin^2(45-a)=1-sin^2(a) is a=360n,a=180+360n,a=1.10714…+180n