Mathematically your statements are incorrect. Let's invetigate one at a time.

a)`costheta/(sin^2theta)=`

`costheta/[sintheta*sintheta]=`

`costheta/sintheta*1/sintheta=`

`cottheta*csctheta`

In general csc of an angle is not equal to the sec of that angle. They are equal if `theta=45+nPi`

b)`cottheta*sectheta=`

`costheta/sintheta*1/costheta=1/sintheta` when `costheta!=0, theta!=Pi/2+nPi`

Here also in general `1/sintheta!=sintheta`

We are given the expression `cot theta*sec theta` :

Expressing these in terms of sin and cos yields:

`costheta/sintheta*1/costheta`

This product is `1/sintheta=csctheta`

**So neither (a) nor (b) are correct simplifications or identities since**

(a) `costheta/(sin^2theta)=cottheta*1/sintheta=cottheta*csctheta` and

(b) `sintheta`

are neither equivalent to `cotthetasectheta` .

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