Solution
sin u + sin q = 27 65 2 sin ( u + q 2 ) . cos ( u − q 2 ) = 27 65 . . . . . . . . . . . . . . . . . . . . . . . . ( 1 ) sin ( u + q ) = 2 sin ( u + q 2 ) . cos ( u + q 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 ) sin ( u + q ) = 2 tan ( u + q 2 ) 1 − tan 2 ( u + q 2 ) = 2 × 3 7 1 − 9 49 = 21 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 ) ( 2 ) = ( 3 ) 2 sin ( u + q 2 ) . cos ( u + q 2 ) = 21 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 4 ) tan ( u + q 2 ) = 3 7 sin ( u + q 2 ) cos ( u + q 2 ) = 3 7 ⟹ cos ( u + q 2 ) = 7 3 sin ( u + q 2 ) . . . . . . . . . . . . ( 5 ) S u b s t i t u t i n g ( 5 ) i n ( 4 ) , 2 sin ( u + q 2 ) . 7 3 . sin ( u + q 2 ) = 21 20 sin 2 ( u + q 2 ) = 9 40 sin ( u + q 2 ) = ± 3 2 10 = + 0.474 o r − 0.474 A c c o r d i n g t o ( 1 ) , W h e n sin ( u + q 2 ) = + 0.474 ⟹ cos ( u − q 2 ) = + 0.438 W h e n sin ( u + q 2 ) = − 0.474 ⟹ cos ( u − q 2 ) = − 0.438 \qquad\qquad
\begin{aligned}
\sin u+\sin q&= \frac{27}{65}\\
2\sin (\frac{u+q}{2}).\cos(\frac{u-q}{2})&= \small \frac{27}{65}........................(1)\\
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\sin(u+q)&=2\sin (\frac{u+q}{2}).\cos(\frac{u+q}{2}).........................(2)\\
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\sin(u+q)&= \frac{2\tan(\frac{u+q}{2})}{1-\tan^2(\frac{u+q}{2})}\\
&= \frac{2\times\frac{3}{7}}{1-\frac{9}{49}}\\
&= \frac{21}{20}.................................(3)\\
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(2)=(3)\\
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2\sin(\frac{u+q}{2}).\cos(\frac{u+q}{2})&= \frac{21}{20}...............................(4)\\
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\tan(\frac{u+q}{2})&= \frac{3}{7}\\
\frac{\sin(\frac{u+q}{2})}{\cos(\frac{u+q}{2})}&= \frac{3}{7}\implies \cos(\frac{u+q}{2})=\frac{7}{3}\sin(\frac{u+q}{2})............(5)\\
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Substituting \,(5)\,in\,(4),
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2\sin(\frac{u+q}{2}).\frac{7}{3}.\sin(\frac{u+q}{2})&=\frac{21}{20}\\
\sin^2(\frac{u+q}{2})&=\frac{9}{40}\\
\sin(\frac{u+q}{2})&=\pm\frac{3}{2\sqrt{10}}\\
&=+0.474\,or\,-0.474\\
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According\,to\,(1),\\
When\,\,\,\sin(\frac{u+q}{2})=+0.474&\implies\cos(\frac{u-q}{2})=+0.438\\
When\,\,\,\sin(\frac{u+q}{2})=-0.474 &\implies\cos(\frac{u-q}{2})=-0.438
\end{aligned} sin u + sin q 2 sin ( 2 u + q ) . cos ( 2 u − q ) sin ( u + q ) sin ( u + q ) ( 2 ) = ( 3 ) 2 sin ( 2 u + q ) . cos ( 2 u + q ) tan ( 2 u + q ) cos ( 2 u + q ) sin ( 2 u + q ) S u b s t i t u t in g ( 5 ) in ( 4 ) , 2 sin ( 2 u + q ) . 3 7 . sin ( 2 u + q ) sin 2 ( 2 u + q ) sin ( 2 u + q ) A ccor d in g t o ( 1 ) , Wh e n sin ( 2 u + q ) = + 0.474 Wh e n sin ( 2 u + q ) = − 0.474 = 65 27 = 65 27 ........................ ( 1 ) = 2 sin ( 2 u + q ) . cos ( 2 u + q ) ......................... ( 2 ) = 1 − tan 2 ( 2 u + q ) 2 tan ( 2 u + q ) = 1 − 49 9 2 × 7 3 = 20 21 ................................. ( 3 ) = 20 21 ............................... ( 4 ) = 7 3 = 7 3 ⟹ cos ( 2 u + q ) = 3 7 sin ( 2 u + q ) ............ ( 5 ) = 20 21 = 40 9 = ± 2 10 3 = + 0.474 or − 0.474 ⟹ cos ( 2 u − q ) = + 0.438 ⟹ cos ( 2 u − q ) = − 0.438
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