Establish the identity. ((cot^2 x)/(csc(x)-1)) = (1+sin(x))/(sin(x)) = ____________________ Could someone explain to me how to solve this?

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Answer 1 · 2018-08-03T22:56:59

Follow proof below.

We have: #frac(cot^(2)(x))(csc(x) - 1)#

One of the Pythagorean identities is #cot^(2)(x) + 1 = csc^(2)(x)#.

We can rearrange it to get:

#Rightarrow cot^(2)(x) = csc^(2)(x) - 1#

Let's apply this rearranged identity to our proof:

#= frac(csc^(2)(x) - 1)(csc(x) - 1)#

The numerator is the difference of two squares, and can be factorised as:

#= frac((csc(x) + 1)(csc(x) - 1))(csc(x) - 1)#

#= csc(x) + 1#

Now, one of the standard trigonometric identities is #csc(x) = frac(1)(sin(x))#.

Applying this, we get:

#= frac(1)(sin(x)) + 1#

#= frac(1 + sin(x))(sin(x)) " " "# #"Q.E.D."#