Question

i.       By differentiating , show that if  θ then .

   ii.       Hence show that

Giving the values of a and b.

  iii.       Find the exact value of

Solution

     i.
 

We are given that;

We are required to show that;

Since   provided that ;

Therefore;

If  and  are functions of , and if , then;

If , then;

Rule for differentiation of  is:

Rule for differentiation of  is;

Since    provided that  and , therefore;

   ii.
 

We are required to show that;

We have found in (i) that;

Second derivative is the derivative of the derivative. If we have derivative
of the curve   as , then expression for the second derivative of the curve  is;

If  and  are functions of , and if , then;

If , then;

Let  and ; then,

Rule for differentiation of  is;

Rule for differentiation of  is;

We have trigonometric identity;

  iii.
 

We are required to find the exact value of;

Rule for integration of  is:

We have trigonometric identity;

Rule for integration of  is:

Rule for differentiation of  is;

Therefore,

  provided that