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3. trignometrical ratios,functions and identities — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS3. trignometrical ratios,functions and identities1 Mark
MCQ
$1 + \cos \,{56^o} + \cos \,{58^o} - \cos {66^o} = $
  • A
    $2\,\cos {28^o}\,\cos \,{29^o}\,\cos \,{33^o}$
  • B
    $4\,\cos {28^o}\,\cos \,{29^o}\,\cos \,{33^o}$
  • ✓
    $4\,\cos {28^o}\,\cos \,{29^o}\,\sin {33^o}$
  • D
    $2\,\cos {28^o}\,\cos \,{29^o}\,\sin \,{33^o}$
✓

Answer

Correct option: C.
$4\,\cos {28^o}\,\cos \,{29^o}\,\sin {33^o}$
c
(c) $1 + \cos 56^\circ + \cos 58^\circ - \cos 66^\circ $

$ = 2{\cos ^2}28^\circ + 2\sin 62^\circ .\sin 4^\circ $

$ = 2{\cos ^2}28^\circ + 2\cos 28^\circ .\sin 4^\circ $

$ = 2\cos 28^\circ (\cos 28^\circ + \cos 86^\circ )$ 

$ = 2\cos 28^\circ .2\cos 57^\circ \cos 29^\circ $

$ = 4\cos 28^\circ \cos 29^\circ \sin 33^\circ $.

Aliter : Apply the conditional identity

$\cos A + \cos B - \cos C = - 1 + 4\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}$                                                              $[\, \because 56^\circ  + 58^\circ  + 66^\circ  = 180^\circ ]$

We get the value of required expression equal to $4\cos 28^\circ \cos 29^\circ \sin 33^\circ $.

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