2tan^2 x-7sec+8=0

2(sin^2 x/cos^2 x)-7cosx+8=0

That's okay except that it should be -7/cos x.

You wanted to multiply all by cos^2 x in order to clear the fraction? Very good! Always clear the fractions first.

Sure you can multiply all by cos^2 x even if "it is all multiplied by 2 i.e. in the brackets".

So, to continue,

2[sin^2(X) /cos^2(X)] -7/cosX +8 = 0

Multiply both sides by cos^2(X),

2sin^2(X) -7cosX +8cos^2(X) = 0

2[1 -cos^2(X)] -7cosX +8cos^2(X) = 0

2 -2cos^2(X) -7cosX +8cos^2(X) = 0

6cos^2(X) -7cosX +2 = 0

Factor that,

(3cosX -2)(2cosX -1) = 0

3cosX -2 = 0

cosX = 2/3

X = arccos(2/3) = 48.1896851 degrees, in the 1st quadrant.

Since cosine is positive also in the 4th quadrant,

X = 360 -48.1896851 = 311.8103149 deg, in the 4th quadrant.

2cosX -1 = 0

cosX = 1/2

X = arccos(1/2)) = 60, in the 1st quadrant.

Since cosine is positive also in the 4th quadrant,

X = 360 -60 = 300 deg, in the 4th quadrant.

Therefore, X = 48.1896851, 60, 300, or 311.8103149 degrees -----answer.

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ALSO: I would be grateful if someone could clarify this for me.

Q. State the values of:

arc sin 0.5 - my answer is 30 degrees

arc tan 1 - my answer is 45 degrees

I ask this because I have no idea what the 'arc' means - does it affect the way I should answer the question as I have never encountered it before.

arcsin(0.5) = 30 deg or 150 deg, since sine is positive in the 1st and 2nd quadrants.

arctan(1) = 45deg or 225deg, since tangent is positive in the 1st and 3rd quadrants.

'arc' here means, or, arcsin(0.5) means an anlge whose sine is 0.5.

arctan(1) is an angle whose tangent is 1.

So associate "arc___" with an angle.