Question

Can someone verify sinx + cosx * cotx = cscx

Answer 1

Hello lfroh.

Answer 2

First of all, welcome to OpenStudy. Nice to see you asking questions here. You'll be now guided to the correct way to get the answer, we will just guide, the rest is up to you!

Answer 3

cotx = 1/tanx

tanx = sinx/cosx

1/tanx = 1/(sinx/cosx) = cosx/sinx

Answer 4

We have : \(\sin x + \cos x \times \cot x = \csc x \) ---( Required to verify)

Answer 5

So now, lfroh. Can you tell me what is \(\cot x\) in terms of \(\cos x\) and \(\sin x\) ?

Answer 6

Yes, cosx/sinx

Answer 7

Great. So I have : \(\cot x = \cfrac{\cos x}{\sin x}\)

Let us put this : in \(\sin x + \cos x \times \cot x \)

Put in the value of \(\cot x\) in the above expression. Can you?

Answer 8

sinx + cosx * cosx/sinx

Answer 9

Excellent. Try to simplify this further, hint : try to cancel out something.

Answer 10

do the sin values cancel out?

Answer 11

Sorry, I did give the hint wrong.

\(\sin x + \cos x \times \cfrac{\cos x}{\sin x}\)

Multiply cos x with cos x , you get?

Answer 12

cos^2x

Answer 13

Right , can you write the whole expression now? In simplified form.

Answer 14

sinx + cos^2x/sinx?

Answer 15

Yes, now : \(\cfrac{\sin x}{1} + \cfrac{\cos^2 x}{\sin x}\)

Try to take LCM , can you?

Answer 16

@lfroh , are you confused anywhere? Let me know, please.

Answer 17

Yes Im not sure where to go from here. Would it be cos^2x/1?

Answer 18

No, do you know how to take LCM?

Answer 19

Well I need to get the same vaule in the denominator but I dont know how.

Answer 20

Right, I want to get \(\sin x\) in the denominator of \(\cfrac{\sin x}{1}\) . To get that, I have to multiply the whole fraction by sin x.

\(\cfrac{\sin x}{1} \times \cfrac{\sin x}{\sin x}\)

mutliply sin x and sin x and : 1 and sin x.

What do you get?

Answer 21

You just have to simplify this first : \(\cfrac{\sin x \times \sin x}{1\times \sin x}\)
@lfroh , where are you stuck?

Answer 22

1/sinx

Answer 23

No, let us solve numerator first. What is sin x * sin x?

Answer 24

sin^2x

Answer 25

great, and in denominator : 1 * sin x =?

Answer 26

sinx, so you would have sin^2x/sinx + sox^2x/sinx

Answer 27

yes! \(\cfrac{\sin^2x}{\sin x} + \cfrac{\cos^2 x}{\sin x}\)

Can you simplify that further?

Answer 28

yes the top simplifies to 1 so it would be 1/sinx. Which equals cscx!

Answer 29

Excellent work. @lfroh , so this is how you should approach to a question , especially of trigonometry. Simplifying is the key to get the answer.

Are you satisfied the help given by me?

Answer 30

Yes! Thank you!

Answer 31

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