Mathematics: Evaluate: 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.

Evaluate: i know that the answer is supposed

Ответ:

zachzach28280

16.01.2022

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

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

$\cot(x+y)=\dfrac{\cos(x+y)}{\sin(x+y)}=\dfrac{\cos x\cos y-\sin x\sin y}{\sin x\cos y+\cos x\sin y}=\dfrac{\cot x\cot y-1}{\cot y+\cot x}$

and so

$\cot\left(x+\dfrac\pi2\right)=\dfrac{\cot x\cot\frac\pi2-1}{\cot\frac\pi2+\cot x}=-\dfrac1{\cot x}=-\tan x$

So the limit is

$\displaystyle\lim_{x\to\frac\pi2^+}\frac{\tan x}{\cot\left(x+\frac\pi2\right)}=\lim_{x\to\frac\pi2^+}\frac{\tan x}{-\tan x}=-1$

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Ответ:

aliceohern

02.03.2022

Mathematics

The answer is 8.8 (100%)

Step-by-step explanation:

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Evaluate:
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.

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Evaluate: 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.

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Evaluate:
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.