# Problems with Trigonometric Identities

We just started this today in class and I have several homework problems, but I don't understand it at all. Can anyone show me step by step how to do this. The problem is $4/(\tan x + \cot x)$.

My teacher went ahead and told us the answer would be $4\sin x\cos x$, so we can make sure we use the right steps and get the correct answer. I am so lost in this. Please help me understand.

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Please elaborate on what to do this this expression. What are the instructions? – King Squirrel May 8 '14 at 21:50
My teacher just said to show the steps using the trig identities to get to the answer.I guess it is considered solving it? Not sure. Sorry – Ila Isabelle May 8 '14 at 21:53

Assuming that your teacher wants the expression simplified:

4/(tanx+cotx)

4/((sin/cosx)+(cosx/sinx))

4/((sin^2x/sinxcosx)+(cos^2x/sinxcosx))

4/((sin^2x+cos^2x)/(sinxcosx))

//sin^2x+cos^2x=1

=> 4/(1/(sinxcosx))

4sinxcosx

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Thank you very much! – Ila Isabelle May 8 '14 at 21:58
The King orders Ila Isabelle to choose his answer by clicking the check mark. And the King says your welcome. – King Squirrel May 8 '14 at 22:00
Lol, as you command King. – Ila Isabelle May 8 '14 at 22:03

Use the definition: $$\tan x=\frac{\sin x}{\cos x},\quad \cot x=\frac{\cos x}{\sin x}.$$ So $$\frac{4}{\tan x+\cot x}=\frac{4}{\dfrac{\sin x}{\cos x}+\dfrac{\cos x}{\sin x}}= \frac{4}{\dfrac{\sin^2 x+\cos^2x}{\sin x\cos x}}$$ Now, what can you do?

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sin^2x + cos^2x= 1, so then I multiply 4*sinx*cosx. Please tell me I am right? – Ila Isabelle May 8 '14 at 21:56
@IlaIsabelle Yes, you are. – egreg May 8 '14 at 21:57
Thank you so much!! – Ila Isabelle May 8 '14 at 21:57