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2015-08-23T21:56:37+05:30

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I am inserting parentheses.. is this right ?

Cos (40 - x) - Sin (50 -x) + (cos² 40 + Cos² 50) / ( Sin² 40  + Sin² 50 )

we know cos A = Sin  (90 - A)
  and  Sin (A+B) - Sin (A-B) = 2 Sin B Cos A

= Sin[90 - (40-x)] - Sin (50-x) + (cos² 40 + Sin² (90 - 50)) /(Sin² 40 + Cos²(90-50)
= Sin (50 +x) - Sin(50 -x) + (cos²40 + Sin²40) /(sin²40 +cos²40)
= 2 Sin x Cos 50 + 1 / 1
= 1 + 2 Sin x Cos 50
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If the question is:
 Cos (40 - x) - Sin (50 -x) + cos² 40 + (Cos² 50 / Sin² 40) + Sin² 50
= [ Sin (90 - 40 + x) + sin (x - 50) ] + cos² 40 + (Cos²50 / Cos²(90-40)) + Cos² (90-50)
= 2 Sin x Cos 50 +  2 Cos² 40 + 1/1
= 2 sin x cos 50 +  (1 + cos 80) + 1
= 2 Sin x  Cos 50 + 2 + Cos (50  + 30)
= 2 Sin x Cos 50 + 2 + √3/2 * cos 50 + 1/2 * Sin 50

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