Jon P. answered 04/28/15
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Knowledgeable Math, Science, SAT, ACT tutor - Harvard honors grad
Let's play with each side and see if they come out the same:
LEFT
tan x + 2 cot x
= tan x + 2 / tan x (since cot x = 1 / tan x)
= (tan2 x + 2) / tan x (using tan x as the common denominator to add the two terms)
= (tan2 x + 1 + 1) / tan x (since 1 + 1 = 2)
= (sec2 x + 1) / tan x (by the pythagorean identity tan2 x = 1 + sec2 x)
= (sec2 x + 1) / (sin x / cos x) (since tan x = sin x / cos x)
= (sec2 x + 1) * (cos x / sin x) (since dividing by a fraction is the same as multiplying by its reciprocal)
= (sec2 x + 1) cos x / sin x (doing the multiplication)
= (sec2 x + 1) cos x / sin x (doing the multiplication)
= (sec2 x cos x + cos x) / sin x (simplifying the numerator)
= (cos x / cos2 x + cos x) / sin x (since sec x = 1 / cos x)
= (1 / cos x + cos x) / sin x (canceling out cos x in the first term)
RIGHT
(sin2 x + 2 cos2 x) / (sin x cos x)
= (sin2 x + cos2 x + cos2 x) / (sin x cos x) (since two of anything is the same as one plus one of it)
= (1 + cos2 x) / (sin x cos x) (by the pythagorean identity sin2 x + cos2 x = 1)
= ((1 + cos2 x) / cos x) / sin x (since dividing by a product is the same as dividing by one term, then
the other)
= (1 / cos x + cos2 x / cos x) / sin x (simplifying the numerator)
= (1 / cos x + cos x) / sin x (canceling out cos x in the second term)
So both sides end up being equal to (1 / cos x + cos x) / sin x. Therefore they are equal to each other.