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Further Trig Problem

y.u.mad.bro?

I got given a problem today in class as a quick finishing question just before the bell rang. I attempted it in class but couldn't finish it.

Unparseable latex formula:

1:\[cosec(x)-tan(x)].

Hence show that

Unparseable latex formula:

\[tan]

Now, my problem is I don't think it can be solved. I discussed with 2 of my mates who also agreed and testing values on a calculator also gave different results.

What I assume the question is supposed to look like is

Unparseable latex formula:

2:\[cosec(x)-cot(x)]

My working so far for the question 1.

Unparseable latex formula:

\[\frac{1}{sin(x)}-\frac{sin(x)}{cos(x)}

Unparseable latex formula:

\frac{cos(x)-sin^2(x)}{sin(x)cos(x)}\]

If anyone has any ideas what I am doing wrong, please let me know :biggrin:

Ideally my tonight because I want to do this question. Also, I just started this topic today in lesson and finished it within two hours so I have't had much practice so any tips would be appreciated :h:

(edited 6 years ago)

Reply 1

RDKGames

Study Forum Helper

Original post by y.u.mad.bro?

...

You're right that that original question is incorrect.

Now you want to find the value of

\tan(15)

. So set

x=30

and your LHS should just be an evaluation of some basic trig ratios.

Reply 2

y.u.mad.bro?

Original post by RDKGames

You're right that that original question is incorrect.

Now you want to find the value of

\tan(15)

. So set

x=30

and your LHS should just be an evaluation of some basic trig ratios.

Thank you. I was starting to feel like an idiot :biggrin:

So, I get the following working out. Hopefully this is right but just want a confirmation.

Unparseable latex formula:

\[cosec(x)-tan(x)\]

Unparseable latex formula:

\frac{1-cos(x)}{sin(x)}\]

Unparseable latex formula:

\[\frac{1-(1-sin^2(\frac{1}{2}x))}{2sin(\frac{1}{2}x)(cos(\frac{1}{2}x))}\]

Unparseable latex formula:

\[\frac{sin\frac{1}{2}x}{cos\frac{1}{2}x}\]

Unparseable latex formula:

\[tan(\frac{1}{2}x)\]

Hence,

Unparseable latex formula:

\[tan(\frac{1}{2}(30))]

Unparseable latex formula:

\[\frac{1-cos(30)}{2sin(\frac{30}{2})}

Unparseable latex formula:

\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}\]

Unparseable latex formula:

\[2-\sqrt{3}\]

Lastly, I know dumb question but how do you derive the identities

Unparseable latex formula:

\[cos(x)=1-2sin^2(\frac{1}{2}x)

sin(x)=2sin(\frac{1}{2}x) cos(\frac{1}{2}x)

Reply 3

RDKGames

Study Forum Helper

Original post by y.u.mad.bro?

Unparseable latex formula:

\[cosec(x)-tan(x)\]

You mean

\csc(x)-\cot(x)

... Not sure why you used

\tan(x)

here.

Also, you were't being asked to derive the answer, simply use the *already given* fact that

\csc(x)-\cot(x)=\tan(\frac{1}{2}x)

by simply plugging in

x=30

into both sides. LHS is just

\frac{1}{\sin(30)}-\frac{\cos(30)}{\sin(30)}

an the RHS is just

\tan(15)

. Express the LHS in terms of ratios, simplify, job done. No identity proving required.

The identities are actually:

\cos(2x) \equiv \cos^2(x)-\sin^2(x)

\sin(2x) \equiv 2\sin(x)\cos(x)

and are given the name 'double angle formulae', and then it's only natural to substitute

\cos^2(x) \equiv 1- \sin^2(x)

into the first one to get

1-2\sin^2(x)

, and then just replace

x

with

\frac{x}{2}

to arrive at the results you used.

In order to prove

\sin(2x), \cos(2x)

identities, you need to work with triangles and/or compound angle formulae.

Reply 4

y.u.mad.bro?

Original post by RDKGames

Sorry my bad. I was still thinking about the first one which was wrong. Thanks for your help anyway. Makes a lot more sense now.

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