cos x csc^2 x + 3 cos x = 7 cos x

Rewriting that,

cosXcsc^2(X) +3cosX = 7cosX

cosXcsc^2(X) +3cosX -7cosX = 0

cosXcsc^2(X) -4cosX = 0

cosX[csc^2(X) -4] = 0

cosX = 0

X = arccos(0) = 90 or 270 degrees

csc^2(X) -4 = 0

csc^2(X) = 4

cscX = +,-2

When cscX = 2,

cscX = 1/sinX

So, 1/sinX = 2

sinX = 1/2 ---positive

Sine is positive in the 1st and 2nd quadrants, so,

X = arcsin(1/2)

X = 30 deg --in the 1st quadrant

X = 180-30 = 150 deg ---in the 2nd quarter

When cscX = -2,

1/sinX = -2

sinX = -1/2 ---negative

Sine is negative in the 3rd and 4th quadrants, so,

X = arcsin(-1/2)

X = 180 +30 = 210 deg --in the 3rd quadrant

X = 360 -30 = 330 deg ---in the 4th quarter

Therefore, if X is from 0 to 360 degrees only,

X = 30, 90, 150, 210, 270, or 330 degrees -------answer.

Check those against the original equation and you'd find them all correct.

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