<s>[INST] How does the trigonometric identity $1 + \cot^2\theta = \csc^2\theta$ derive from the identity $\sin^2\theta + \cos^2\theta = 1$? I would like to understand how would the original identity of $$ \sin^2 \theta + \cos^2 \theta = 1$$ derives into 
$$  1 + \cot^2 \theta = \csc^2 \theta  $$
This is my working:
a) $$ \frac{\sin^2 \theta}{ \sin^2 \theta } + \frac{\cos^2 \theta }{ \sin^2 \theta }= \frac 1  { \sin^2 \theta } $$
b) $$1 + \cot^2 \theta = \csc^2 \theta
$$
How did the $ \tan^2 \theta + 1 = \sec^2 \theta$ comes into the picture?


 [/INST] It is taken from Pythagoras theorem,
Let 
Know that
let 
c= Hypothenuse 
a= Opposite
b=Adjacent 
length of the triangle can be found by
$$a^2+b^2=c^2$$
Now divide everything by b
We have
$$\frac{a^2}{b^2}+1=\frac{c^2}{b^2}$$
Notice that 
$$tan \theta=\frac{a}{b}$$
$$sec \theta=\frac{1}{\cos \theta}=\frac{c}{b}$$
There you go 
$$1+\tan^2 \theta=\sec^2 \theta$$
There rest of the two identities are obtained by dividing c&a!
Which is 
Division by c
$$\sin^2 \theta+\cos^2 \theta=1$$
Division by a
$$1+\cot^2\theta=\csc^2\theta$$
</s>