Question

If sec theta minus tan theta is equal to 4,then find the value of cosec theta.

Answer 1

Sec theta - tan theta = 4
(1-Sin theta)/Cos theta = 4
Multiply and divide by (1+Sin theta)
Thus 
   1-Sin^2 theta / Cos theta (1+Sin the ta) = 4
    Cos^2 theta/Cos theta(1+Sin theta) = 4
       Cos theta/ (1+Sin theta) = 4
     Root (1-Sin^2 theta) / 1+Sin theta = 4
 
Squaring both Sides
  
            1-Sin^2 theta / (1+Sin theta)^2 = 16
             (1-Sin theta)(1+Sin theta) / (1+Sin theta)^2 = 16
                   1-Sin theta / 1+Sin theta = 16
                     1-Sin theta = 16 + 16 Sin theta
                        17 Sin theta = -15
                           Sin theta = -15/17
 Thus                   Cosec theta = -17/15

Answer 2

Given  SecФ - Tan Ф = 4       ----(1)
Multiply both sides by  SecФ + TanФ to get:

     Sec²Ф - Tan²Ф = 4 (SecФ + TanФ)
=> SecФ + TanФ = 1/4     ---- (2)

From (1) & (2),   SecФ = 17/8   and  Tan Ф = -15/8
So   SinФ = - 15/17

=>  CosecФ = - 17/15            negative real number.