QR Destroyer #146

[txchexmex]

Could someone please help me with QR Destroyer #146.

f(x)= tan x + cot x
g(x) = csc x
find f(x) divided by g(x)?

In the answer explanation, I am not sure how they placed cos x as the denominator after multiplying through by the LCD. If the answer its obvious, I am just not seeing it and probably just need to take a break.

TIA

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[Shinpe]

I don't know how it's done in the solution but this is how I do it. I usually start by just converting everything to Sin (S) and Cos (C) unless I see a shorter way.

(S/C + C/S)/(1/S) = S*(S/C + C/S) = S^2/C + C (common denominator is C since the second doesnt have a denominator) = (S^2 + C^2)/C = 1/C= Sec(x)

That looks kinda annoying to read through but I hope it helps.