Prove that-i 2 sin5 -12sin-12-1-2ii 2 cos5 -12cos-12iii 2 sin5 -12cos5 -12-3-2-2

(i) 2 sin 5π/12 sin π/12=1/2 ∵ 2 sin A sin B =cos(A B) cos (A+B) ⇒ 2 sin 5π/12 sin π/12 =cos ( 5π/12 π/12 ) cos ( 5π/12+ π/12 ) =cos ( 4π/12 ) cos ( 6π ...

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Byju's Answer

Standard XII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

Prove that:i ...

Question

Prove that:
(i) $2 s i n \frac{5 π}{12} s i n \frac{π}{12} = \frac{1}{2}$
(ii) $2 c o s \frac{5 π}{12} c o s \frac{π}{12}$
(iii) $2 s i n \frac{5 π}{12} c o s \frac{5 π}{12} = \frac{\sqrt{3} + 2}{2}$

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Solution

(i) $2 s i n \frac{5 π}{12} s i n \frac{π}{12} = \frac{1}{2} ∵ 2 s i n A s i n B = c o s (A - B) - c o s (A + B) \Rightarrow 2 s i n \frac{5 π}{12} s i n \frac{π}{12} = c o s (\frac{5 π}{12} - \frac{π}{12}) - c o s (\frac{5 π}{12} + \frac{π}{12}) = c o s (\frac{4 π}{12}) - c o s (\frac{6 π}{12}) = c o s (\frac{π}{3}) - c o s (\frac{π}{2}) = \frac{1}{2} - 0 = \frac{1}{2} R H S$

(ii) $2 c o s \frac{5 π}{12} c o s \frac{π}{12} = \frac{1}{2} ∵ 2 c o s A c o s B = c o s (A + B) + c o s (A - B) \Rightarrow 2 s i n \frac{5 π}{12} s i n \frac{π}{12} = c o s (\frac{5 π}{12} + \frac{π}{12}) + c o s (\frac{5 π}{12} - \frac{π}{12}) = c o s (\frac{π}{2}) + c o s (\frac{π}{3}) = 0 + \frac{1}{2} = \frac{1}{2} = R H S$

(iii) $2 s i n \frac{5 π}{12} c o s \frac{π}{12} = \frac{\sqrt{3} + 2}{2} ∵ 2 s i n A c o s B = s i n (A + B) + s i n (A - B) = (\frac{5 π}{12} + \frac{π}{12}) + s i n (\frac{5 π}{12} - \frac{π}{12}) = s i n \frac{π}{2} + \frac{π}{3} = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2} = R H S$
(Taking LCM)

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Similar questions

2 sin (\frac{5 π}{12}) sin (\frac{π}{12}) =

Q. Prove that:
(i)

2 \sin \frac{5 π}{12} \sin \frac{π}{12} = \frac{1}{2}

(ii)

2 \cos \frac{5 π}{12} \cos \frac{π}{12} = \frac{1}{2}

(iii)

2 \sin \frac{5 π}{12} \cos \frac{π}{12} = \frac{\sqrt{3} + 2}{2}

Q. Let

z_{1} = 1 - i

and

z_{2} = \sqrt{3} + i

, then

z_{1} \cdot z_{2}

in polar form is

Show that:
(i) $s i n 50^{\circ} c o s 85^{\circ} = \frac{1 - \sqrt{2} s i n 35^{\circ}}{2 \sqrt{2}}$
(ii) $s i n 25^{\circ} c o s 115^{\circ} = \frac{1}{2} (s i n 140^{\circ} - 1)$

Q. Numerical value of

cos \frac{π}{12} (sin \frac{5 π}{12} + cos \frac{π}{4}) + sin \frac{π}{12} (cos \frac{5 π}{12} - sin \frac{π}{4})

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Prove that: (i) 2 sin 5π12 sinπ12=12 (ii) 2 cos 5π12 cosπ12 (iii) 2 sin 5π12 cos 5π12=√3+22

Prove that:
(i) $2 s i n \frac{5 π}{12} s i n \frac{π}{12} = \frac{1}{2}$
(ii) $2 c o s \frac{5 π}{12} c o s \frac{π}{12}$
(iii) $2 s i n \frac{5 π}{12} c o s \frac{5 π}{12} = \frac{\sqrt{3} + 2}{2}$