What is the general solution for cos(4y)=2cos(2y)-1 ?

The general solution for cos(4y)=2cos(2y)-1 is y=pin,y= pi/4+pin,y=(3pi)/4+pin

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cos(4y)=2cos(2y)-1

cos (4 y) = 2 cos (2 y) - 1

y = πn, y = \frac{π}{4} + πn, y = \frac{3 π}{4} + πn

Solution steps

cos (4 y) = 2 cos (2 y) - 1

Subtract

2 cos (2 y) - 1

from both sides

cos (4 y) - 2 cos (2 y) + 1 = 0

Rewrite using trig identities

- 2 cos (2 y) + 2 cos^{2} (2 y) = 0

Solve by substitution

cos (2 y) = 1, cos (2 y) = 0

cos (2 y) = 1 : y = πn

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

Combine all the solutions

y = πn, y = \frac{π}{4} + πn, y = \frac{3 π}{4} + πn

csc (x) cot (x) = 23

sin (θ) cos (θ) - sin (θ) = 0

cos^{2} (θ) + 2 sin (θ) + 1 = 0

arcsec (x) = arcsec (2)

tan^{2} (x) + (3 - 1) tan (x) - 3 = 0

What is the general solution for cos(4y)=2cos(2y)-1 ?
The general solution for cos(4y)=2cos(2y)-1 is y=pin,y= pi/4+pin,y=(3pi)/4+pin