Question

     i.       By differentiating  , show that if y = cot x then

   ii.       Hence, show that

 

By using appropriate trigonometrical identities, find the exact value of

 

  iii.

 

  iv.

 

Solution

     i.
 

We are given;

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is:

If  and  are functions of , and if , then;

Let  and , then;

Rule for differentiation of  is;

Rule for differentiation of  is;

We have the trigonometric identity;

   provided that

Therefore, if;

   provided that

Hence;

For;

As demonstrated above;

Hence;

 

   ii.
 

As we have demonstrated in (i) that;

Therefore, the inverse of differentiate ie integral must be as;

Hence;

   provided that

 

  iii.
 

Utilizing the identity;

We can write;

Rule for integration of  is:

As demonstrated in (ii);

Hence;

Rule for integration of  is:

 

  iv.
 

We have the trigonometric identity;

Rule for integration of  is:

   provided that

We have shown in (ii) that;

Therefore;