Question

cosec theta -cot theta =1/4 ,then find the value of cosec theta +cot theta

Answer 1

According to question,
                          Cosec theta - Cot theta = 1/4
Expanding it into cos and sin

                         1/Sin theta - Cos theta / Sin theta = 1/4
taking the denominator,
W get, 
                           (1 - Cos theta) / Sin theta = 1/4
Now doing the reciprocal on both the sides

We get,
                         (Sin theta) / (1-Cos theta)  = 4
Multiplying the numerator and the denominator of LHS with (1 +Cos theta)
 we get,
                 Sin theta * (1 + Cos theta) / (1 - Cos theta) * (1 + Cos theta) = 4
We know,
              (a+b) (a-b)  = a^2 -b^2
So (1 + Cos theta) (1- Cos theta) = 1- Cos^2 theta
                                                     =  Sin^ theta
Thus putting this term in the above equation,
 
                  Sin theta * (1 + Cos theta) / Sin^2 theta = 4
                                    (1 + Cos theta) / Sin theta = 4
Separating the denominator,
                           1/Sin theta + Cos theta / Sin theta = 4
                        Cosec theta +  Cot theta = 4
Thus, the value of Cosec theta + Cot theta is 4