prove cotx-tanx=cos2x/sinxcosx?

Question

i don't know how to do this...i've tried several techniques..but none seem to be working..

Answer 1

As I mentioned in another post here the other day, there is one technique that *always* works, even if it isn't the fastest.

Change everything to sin and cos. Use sin^2 x + cos^2 x = 1. Simplify. Thats it.

So, the LHS is
1/tan x - tan x
= cos x / sin x - sin x / cos x
= (cos^2 x - sin^2 x) / (sin x cos x).
Now, we have the right denominator, but the numerator isn't right yet. What else can we simplify? We know how to change cos2x into something just involving cos x or sin x:
You may already know that cos 2x = cos^2 x - sin^2 x. If not, maybe you know this one:
cos 2x = 2 cos^2 x - 1
= 2 cos^2 x - (sin^2 x + cos^2 x)
= cos^2 x - sin^2 x.
And thus the two sides are equivalent.

Answer 2

Since cos2x = cos^2 x - sin^2 x is a basic identity,
the right hand side becomes (cos^2 x - sin^2 x)/sinxcosx.

Separating terms this reduces to cosx/sinx - sinx/cosx.

But this equals cotx - tanx by definition of tanx and cotx.

Answer 3

cos x+a million/cot=sinx+tanx you'll adjust area one to equivalent area 2. First replace the cotx into cosx/sinx: cosx+a million/ cosx/sinx= sinx + tanx Now, turn the denominator and multiply it with the help of the numberator, so that you cancel the sins contained in the first time period, and get cosx: cosx+a million/ (sinx/cosx)= sinx + tanx Now your issue appears like this, so all you want to do is multiply the a million with the help of sinx/cosx, it rather is likewise stated as tanx, and as all of us comprehend something situations a million is the unique time period, so that you're left with tanx: sinx +a million(sinx/cosx)= sinx + tanx Now you're performed, and look at were given: sinx + tanx = sinx + tanx

Answer 4

Turn the left side into the right by converting everything into sines and cosines. I assume you know your basic trig identities?