Prove that-i sin- 3-x cos- 6-x-cos- 3-x sin- 6-x-1ii sin 4 - 9-7 cos- 9-7-cos 4 - 9-7 sin- 9-7-32iii sin 3 - 8-5 cos- 8-5-cos 3 - 8-5 sin- 8-5-1

(i) #960;3=60 #176;, #160; #960;6=30 #176; LHS #160;= #160;sin60 #176; x #160;cos30 #176;+x #160;+cos60 #176; x #160;sin30 #176;+x ...

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Standard XII

Mathematics

Chain Rule of Differentiation

Prove that:i ...

Question

Prove that:
(i) $\sin (\frac{π}{3} - x) \cos (\frac{π}{6} + x) + \cos (\frac{π}{3} - x) \sin (\frac{π}{6} + x) = 1$
(ii) $\sin (\frac{4 π}{9} + 7) \cos (\frac{π}{9} + 7) - \cos (\frac{4 π}{9} + 7) \sin (\frac{π}{9} + 7) = \frac{\sqrt{3}}{2}$
(iii) $\sin (\frac{3 π}{8} - 5) \cos (\frac{π}{8} + 5) + \cos (\frac{3 π}{8} - 5) \sin (\frac{π}{8} + 5) = 1$

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Solution

(i) $\frac{π}{3} = 60 °, \frac{π}{6} = 30 °$
$LHS = \sin (60 ° - x) \cos (30 ° + x) + \cos (60 ° - x) \sin (30 ° + x) = \sin [(60 ° - x) + (30 ° + x)] (U sing the formula \sin A \cos B + \cos A \sin B = \sin (A + B) and taking A = 60 ° - x a n d B = 30 ° + x) = \sin 90 ° = 1 = RHS Hence proved .$

(ii)
$\sin (\frac{4 π}{9} + 7) \cos (\frac{π}{9} + 7) - \cos (\frac{4 π}{9} + 7) \sin (\frac{π}{9} + 7) = \sin [(\frac{4 π}{9} + 7) - (\frac{π}{9} + 7)] [\sin A \cos B - \cos A \sin B = \sin (A - B)] = \sin \frac{3 π}{9} = \sin \frac{π}{3} = \frac{\sqrt{3}}{2}$

(iii)
$\sin (\frac{3 π}{8} - 5) \cos (\frac{π}{8} + 5) + \cos (\frac{3 π}{8} - 5) \sin (\frac{π}{8} + 5) = \sin [(\frac{3 π}{8} - 5) + (\frac{π}{8} + 5)] [\sin A \cos B + \cos A \sin B = \sin (A + B)] = \sin \frac{4 π}{8} = \sin \frac{π}{2} = 1$

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Similar questions

Q. Prove that:
(i) sin (60° − θ) cos (30° + θ) + cos (60° − θ) sin (30° + θ) = 1.
(ii)

\sin (\frac{4 π}{9} + 7) \cos (\frac{π}{9} + 7) - \cos (\frac{4 π}{9} + 7) \sin (\frac{π}{9} + 7) = \frac{\sqrt{3}}{2}

(iii)

\sin (\frac{3 π}{8} - 5) \cos (\frac{π}{8} + 5) + \cos (\frac{3 π}{8} - 5) \sin (\frac{π}{8} + 5) = 1

Prove that:
(i) $\frac{1}{3 + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + 1} = 1$
(ii) $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} = 2$

Prove that:
(i) $s i n (60^{\circ} - θ) c o s (30^{\circ} + θ) + c o s (60^{\circ} - θ) s i n (30^{\circ} + θ) = 1$
(ii) $s i n (\frac{4 π}{7} + 7) c o s (\frac{π}{9} + 7) - c o s (\frac{4 π}{9} + 7) s i n (\frac{π}{9} + 7) = \frac{\sqrt{3}}{2}$
(iii) $s i n (\frac{3 π}{8} - 5) c o s (\frac{π}{8} + 5) + c o s (\frac{3 π}{8} - 5) s i n (\frac{π}{8} + 5) = 1$

Q. If

\frac{2}{x + 2 y} + \frac{1}{x - 2 y} = \frac{- 5}{7}

then prove that

\frac{x}{y} = \frac{2 (10 y - 7)}{(5 x - 21)}

Q. Prove that:

(i)

{(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}} = 1

(ii)

{(\frac{x^{a}}{x^{b}})}^{c} \times {(\frac{x^{b}}{x^{c}})}^{a} \times {(\frac{x^{c}}{x^{a}})}^{b} = 1

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