Question

Evaluate:
`\lim_{x \to \ ( \pi)/(2)^+ } (tan(x))/(cot(x+ (\pi)/(2)))`

I know that the answer is supposed to be -1, but I'm unsure of how to get it. A hint says to use trig identities cleverly, as L'Hopital's rule will catch you in a loop.

Answer 1

The trig identity in question is
`\cot\left(x+\frac\pi2\right)=-\tan x`.

This follows from the angle sum identities for cosine and sine.


`\cot(x+y)=(\cos(x+y))/(\sin(x+y))=(\cos x\cos y-\sin x\sin y)/(\sin x\cos y+\cos x\sin y)=(\cot x\cot y-1)/(\cot y+\cot x)`

and so


`\cot\left(x+\frac\pi2\right)=(\cot x\cot\frac\pi2-1)/(\cot\frac\pi2+\cot x)=-\frac1{\cot x}=-\tan x`

So the limit is


`\displaystyle\lim_(x\to\frac\pi2^+)(\tan x)/(\cot\left(x+\frac\pi2\right))=\lim_(x\to\frac\pi2^+)(\tan x)/(-\tan x)=-1`