Given-sin-2x-sin-2y-5-12-cos-x-y-2-3-sin-x-y-where-0-lt-x-y-lt-pi-find-the-value-of-cos-x-y-2sin-x-y-

Question Number 105861 by bemath last updated on 01/Aug/20

G i v e n {\begin{cases} \sin 2 x - \sin 2 y = - \frac{5}{12} \\ \cos (x + y) = - \frac{2}{3} \sin (x - y) \end{cases}

w h e r e 0 < x - y < π .

f i n d t h e v a l u e o f \cos (x + y) + 2 \sin (x - y)

Commented by PRITHWISH SEN 2 last updated on 01/Aug/20

\sin 2 x - \sin 2 y = - \frac{5}{12}

\Rightarrow 24 \sin (x - y) \cos (x + y) = - 5 \dots \dots (i)

3 \cos (x + y) + 2 \sin (x - y) = 0 \dots \dots . (ii)

3 \cos (x + y) - 2 \sin (x - y) = \sqrt{{3 \cos (x + y) + 2 \sin (x - y)}^{2} - 24 \sin (x - y) \cos (x + y)}

= \pm \sqrt{5} \dots . . (iii)

from (ii) - (iii)

4 \sin (x - y) = \pm \sqrt{5}

Again given

\cos (x + y) = - \frac{2}{3} \sin (x - y)

\Rightarrow \frac{\cos (x + y)}{2 \sin (x - y)} + 1 = 1 - \frac{1}{3}

\Rightarrow \cos (x + y) + 2 \sin (x - y) = \frac{4 \sin (x - y)}{3} = \pm \frac{\sqrt{5}}{3}

Answered by Dwaipayan Shikari last updated on 01/Aug/20

s i n 2 x - s i n 2 y = - \frac{5}{12}

2 c o s (x + y) s i n (x - y) = - \frac{5}{12}

- \frac{4}{3} {s i n}^{2} (x - y) = - \frac{5}{12}

s i n (x - y) = \pm \frac{\sqrt{5}}{4}

s o c o s (x + y) = \pm \frac{\sqrt{5}}{6}

c o s (x + y) + 2 s i n (x - y) = \pm (- \frac{\sqrt{5}}{6} + \frac{\sqrt{5}}{2}) = \pm \frac{\sqrt{5}}{3}