If-for-0-lt-x-lt-pi-2-y-exp-sin-2-x-sin-4-x-sin-6-x-log-e-2-is-a-zero-of-the-quadratic-equation-x-2-9x-8-0-then-the-value-of-sin-x-cos-x-sin-x-cos-x-is-

Question Number 56415 by gunawan last updated on 16/Mar/19

If for 0 < x < \frac{π}{2},

y = e x p [(\sin^{2} x + \sin^{4} x + \sin^{6} x + \dots \infty) \log_{e} 2]

is a zero of the quadratic equation

x^{2} - 9 x + 8 = 0, then the value of \frac{\sin x + \cos x}{\sin x - \cos x} is

Answered by tanmay.chaudhury50@gmail.com last updated on 16/Mar/19

y = e^{[({s i n}^{2} x + {s i n}^{4} x + \dots \infty) l n 2]}

n o w s = {s i n}^{2} x + {s i n}^{4} x + \dots \infty

s = \frac{{s i n}^{2} x}{1 - {s i n}^{2} x} = {t a n}^{2} x

y = e^{{t a n}^{2} x \times l n 2}

x^{2} - 9 x + 8 = 0

(x - 1) (x - 8) = 0

s o v a l u e o f y i s e i t h e r = 1 o r 8

1 = e^{{t a n}^{2} x \times l n 2}

e^{0} = e^{{t a n}^{2} x \times l n 2}

\boldsymbol b u t \boldsymbol t a n x \neq 0

s o 8 = e^{{t a n}^{2} x \times l n 2}

{t a n}^{2} x \times l n 2 = l n 8

{t a n}^{2} x \times l n 2 = 3 l n 2

t a n x = \sqrt{3}

\frac{s i n x + c o s x}{s i n x - c o s x}

= \frac{t a n x + 1}{t a n x - 1} \to \frac{\sqrt{3} + 1}{\sqrt{3} - 1}

\frac{3 + 2 \sqrt{3} + 1}{3 - 1}

= \frac{4 + 2 \sqrt{3}}{2}

= 2 + \sqrt{3} \boldsymbol a n s w e r