Originally Posted by

**ticbol** 1)Factor & simplify: Tan^(3)x+8

That could be a typo.

I see that it should be tan^(3)x = 8

Then, take the cube roots of both sides,

tanX = 2 ......a positive tan value, so X must be in the 1st or 3rd quadrants.

X = arctan(2)

X = 1.10715 radians, in the 1st quadrant

X = (pi +1.010715) = 4.24874 radians, in the 3rd quadrant

So,

X = 1.10715 and 4.24874 radians

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2) Sec^(2)x-2Tan^(2)x = 0

In the trig identity

sin^2(X) +cos^2(X) = 1,

Divide both sides by cos^2(X),

tan^2(X) +1 = sec^2(X) -------**

So,

tan^2(X) +1 -2tan^2(X) = 0

-tan^2(X) = -1

tan^2(X) = 1

tanX = +,-1

When tanX = 1,

positive tan value, so X is in the 1st and 3rd quadrants,

X = arctan(1)

X = pi/4 and (pi +pi/4) = 5pi/4

or, X = pi/4, and 5pi/4

When tanX = -1,

negative tan value, so X is in the 2nd and 4th quadrants,

X = arctan(-1)

X = (pi -pi/4) = 3pi/4 in the 2nd quadrant

X = (2pi -pi/4) = 7pi/4 in the 4th quadrant

Therefore, X = pi/4, 3pi/4, 5pi/4 and 7pi/4

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3) Tanx = -sqrt(3)/3

If you divide both numerator and denominator by sqrt(3), that's the same as

tanX = -1 / sqrt(3)

Negative tan, so X is in the 2nd and 4th quadrants.

X = arctan[-1 / sqrt(3)]

X = (pi -pi/6) = 5pi/6 in the 2nd quadrant

X = (2pi -pi/6) = 11pi/6 in the 4th quadrant

So,

X = 5pi/6 and 11pi/6