Verify the pythagorean identity.

Convert  to sines and cosines:-
 1 + cot^2 x
=  1 + cos^2 x / sin^2x

=  sin^2 x + cos^2 x
   
              sin^2 x

but sin^2 x + cos^2 x = 1

so this =  1 / sin^2x  which is the same as csc^2 x

1 + cot² t = csc² t
[tex]1+cot ^{2} t = \frac{1}{sin ^{2} t} \\ 1+ \frac{cos^{2}t }{sin^{2}t } = \frac{1}{sin ^{2} t}[/tex]  / * sin² t
sin² t + cos² t = 1
1 = 1
We have confirmed the identity.

The identity is verified. See the explanation.

Step-by-step explanation:

You must keep on mind the following identities:

[tex]csc\theta=\frac{cos\theta}{sin\theta}\\\\csc\theta=\frac{1}{sin\theta}\\\\sin^2\theta+cos^2\theta=1[/tex]

Therefore, by substitution, you can rewrite the identity as shown below:

[tex]1+cot^2\theta=csc^2\theta\\\\1+\frac{cos^2\theta}{sin^2\theta}=csc^2\theta[/tex]

Simpliying, you obtain:

[tex]\frac{sin^2\theta+cos^2\theta}{sin^2\theta}=csc^2\theta\\\\\frac{1}{sin^2\theta}=csc^2\theta\\\\csc^2\theta=csc^2\theta[/tex]

The identity is verified.

To prove this Pythagorean Identity, we have to know that:

cot = 1 / tan

and we also know that:

tan = sin / cos

Therefore,

cot = 1 / tan = cos / sin

So we can write the given equation in the form of:

1 + (cos^2 θ / sin^2 θ) = csc^2 θ

Expanding the left hand side of the equation:

(sin^2 θ / sin^2 θ) + (cos^2 θ / sin^2 θ) = csc^2 θ

(sin^2 θ + cos^2 θ) / sin^2 θ = csc^2 θ

We know that given a unit circle, sin^2 θ + cos^2 θ = 1. So:

1 / sin^2 θ = csc^2 θ

The equation above is already true basing on the trigonometric identities. Therefore:

csc^2 θ = csc^2 θ

This is pretty simple.

If you have a sheet that tells you your identities then you should see that 
[tex]1+cot^{2}=csc^{2}[/tex]
Just change
[tex]1+cot^{2}[/tex] to [tex]csc^{2}[/tex]
and get 
[tex]csc^{2}=csc^{2}[/tex]