Question

cot 3 theta = 4/3 where theta is less than pie/6 then the value of 3cosectheta-4sectheta is

Answer 1

The value of $3\csc \theta -4\sec \theta$ is 10.

Step-by-step explanation:

1. Given $\cot 3\theta =\frac{4}{3}$

  So $\sin 3\theta =\frac{3}{5}$        ...1)

   and $\cos 3\theta =\frac{4}{5}$     ....2)

2. From question

    $3\csc \theta -4\sec \theta$      

   which can be written as

  $\frac{3}{\sin \theta } - \frac{4}{\cos \theta }$      ...3)

3.  Now again equation 3) can be written as

   $\frac{3\cos \theta -4\sin \theta }{\cos \theta\sin \theta}$

   above equation can also be written as

   $\frac{5\times (\frac{3}{5}\cos \theta -\frac{4}{5}\sin \theta) }{\cos \theta\sin \theta}$       ...4)

4. Now from equation 1), equation 2) and equation 4)

    $\frac{5\times (\sin 3\theta \cos \theta -\cos 3\theta \sin \theta) }{\cos \theta\sin \theta}$     ...5)

5.   Now equation 5) can be written as

     $\frac{5\times \sin (3\theta -\theta ) }{\cos \theta\sin \theta}$

    $\frac{5\times \sin 2\theta}{\cos \theta\sin \theta}$   ...6)

6.  Now from equation 6)

   $\frac{5\times 2\sin \theta \cos \theta}{\cos \theta\sin \theta}$ = 10