Establish each identity

Question

csc * cos = cot 

and

1-(sin^2/1+cos) = cos 

Answer 1

(csc x)(cos x) = (1/sin x)(cos x) = (cos x) / (sin x) = cot x

and

1 - (sin² x)/(1 + cos x)

= 1 - (sin² x)(1 - cos x) / [(1 + cos x)(1 - cos x)]

= 1 - (sin² x)(1 - cos x) / (1 - cos² x)

1 - (sin² x)(1 - cos x) / sin² x

= 1 - (1 - cos x)

= cos x

Answer 2

Hi Adin,

We can establishing new Trig identities by making use of the known Trig identities.

Let's apply this approach with our first Trig identity, we need to prove:  csc * cos = cot

We can replace the csc on the left side of the equation with the known reciprocal identity for csc:

csc = 1 / sin

Plugging 1 / sin in for csc, our equation becomes:  (1 / sin) * cos = cot

Now we can simplify the left side of the equation by doing the multiplication and we get:  cos / sin = cot

Lastly, we can replace the Trig ratio (cos / sin) on the left side of the equation with its equivalent single
Trig function (cot):  cot = cot

Our identity is now proved, since both sides of our equation match.

Let's now use the same substitution approach to prove our 2nd identity: 

1 - (sin^2 / 1 + cos) = cos

We can start by deriving an equivalent expression for sin^2 in terms of cos^2, so that both sides of our equation use only the cos function.  We can do this by making use of the Pythagorean identity:

sin^2 + cos^2 = 1, which gives us:  sin^2 = 1 - cos^2

Now let's plug in our derived expression for sin^2 in terms of cos^2:

1 - (1 - cos^2) / 1 + cos) = cos

Factoring the numerator of the fraction on the left side of the equation, we get:

1 - ( (1 + cos) (1 - cos) / (1 + cos) ) = cos

Cancelling the common factor (1 + cos) from both the numerator and denominator in the fraction on the left side of the equation, we get:

1 - ( (1 + cos) (1 - cos) / (1 + cos) ) = cos

1 - (1 - cos) = cos

Distributing the minus sign over each term in parens on the left side of the equation, we get:

1 - 1 + cos = cos

The 1's cancel out (1 - 1 = 0) on the left side of the equation, leaving us with our desired result:
cos = cos

I trust the above will be helpful to you in being able to use the same kind of substitution techniques in the proving out of other Trig identities.

Thanks for submitting your question.

Regards, Jordan.