The-sum-of-all-the-solutions-of-the-equation-1-2-cosec-x-sec-2-x-2-2-in-the-interval-0-4pi-is-npi-where-n-is-equal-to-

Question Number 22348 by Tinkutara last updated on 16/Oct/17

The sum of all the solutions of the

equation 1 + 2 cosec x = - \frac{\sec^{2} \frac{x}{2}}{2} in

the interval [0, 4 π] is n π, where n is

equal to

Answered by ajfour last updated on 16/Oct/17

n = 5, a s t h e r e a r e t w o

p o s s i b l e s o l u t i o n s i n t h e g i v e n

i n t e r v a l . x = \frac{3 π}{2}, \frac{7 π}{2} .

L e t \tan \frac{x}{2} = t

2 cosec x = \frac{2}{\sin x} = \frac{1 + t^{2}}{t}

\sec^{2} \frac{x}{2} = 1 + t^{2}; T h e n

1 + \frac{1 + t^{2}}{t} = - \frac{(1 + t^{2})}{2}

2 t + 2 + 2 t^{2} + t (1 + t^{2}) = 0

o r t^{3} + 2 t^{2} + 3 t + 2 = 0

\Rightarrow (t + 1) (t^{2} + t + 2) = 0

o n l y o n e r e a l r o o t t = - 1

\Rightarrow \tan (\frac{x}{2}) = - 1

A s 0 ⩽ \frac{x}{2} ⩽ 2 π

\Rightarrow \frac{x}{2} = \frac{3 π}{4}, \frac{7 π}{4}

\Rightarrow x = \frac{3 π}{2}, \frac{7 π}{2}

T h e i r s u m = 5 π = n π

S o n = 5 .

Commented by Tinkutara last updated on 16/Oct/17

Thank you very much Sir!