Question

Find the value of  $$ \dfrac{\sin^6 15^\circ + \sin^6 75^\circ + 6 \sin^2 15^\circ + \sin^2 75^\circ}{\sin^4 15^\circ + \sin^4 75^\circ + 5 \sin^2 15^\circ \sin^2 75^\circ} $$ 

• A. sin 15°+sin 75°
• B. 6/5
• C. 1
• D. sin 15°+cos 15°
• E. None of the above

Answer 1

The Correct Option is C

Let a = sin2 15°
b = sin2 75° = sin2 (90° - 15°) = cos2 15°
. So, a + b = 1.
Thus,
sin6 15° + sin6 75° + 6sin2 15° + sin2 75°/sin4 15° + sin4 75° + 5sin2 15° sin2 75° =a3+b3+6ab/a2+b2+5ab
= (a+b)(a2+b2-ab) + 6ab/(a+b)2 + 3ab
= (a+b)((a+b)2 - 3ab) + 6ab/(a+b)2 + 3ab using a + b = 1
= -3ab +6ab/3ab
= 1
Hence, option C is the correct answer.