Question

sin*sin6x-sin*sin4x=sin2xsin10x prove

Answer 1

$\large\underline\blue{\bold{Given \:  Question :-  }}$

$\bf \:  {sin}^{2} 6x - {sinx}^{2} 4x = sin2x \: sin10x$

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$\bf \:\huge \red{AηsωeR} ✍$

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$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$\bf \:  1. \: {sin}^{2} x = \dfrac{1 - cos2x}{2}$

$\bf \:  2. \: cos \: x \: - cos \: y = - 2sin \bigg(\dfrac{x + y}{2} \bigg)\bigg( \dfrac{x - y}{2} \bigg)$

$\bf \:  3. \: sin( - x) = - sin \: x$

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$\large\underline\purple{\bold{Solution :- }}$

$\sf \:  ⟼{sin}^{2} 6x - {sinx}^{2} 4x$

Using above identity (1), we get

$\sf \:  ⟼\bigg( \dfrac{1 - cos(2 \times 6x)}{2} \bigg) - \bigg( \dfrac{1 - \: cos(2 \times 4x) }{2} \bigg)$

$\sf \:  ⟼\dfrac{1 - cos12x - 1 + cos8x}{2}$

$\sf \:  ⟼\dfrac{cos8x - cos12x}{2}$

$\sf \:  ⟼\dfrac{1}{ \cancel2} \bigg( - \cancel2 \: sin \: (\dfrac{12x + 8x}{2} ) \: sin \: (\dfrac{8x - 12x}{2}) \bigg)$

$\sf \:  ⟼ \: - \: sin(10x) \: sin( - \: 2x)$

$\sf \:  ⟼ \: sin \: (10x) \: sin \: (2x)$

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$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

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Trigonometry Formulas

sin(−θ) = −sin θ

cos(−θ) = cos θ

tan(−θ) = −tan θ

cosec(−θ) = −cosecθ

sec(−θ) = sec θ

cot(−θ) = −cot θ

Product to Sum Formulas

sin x sin y = 1/2 [cos(x–y) − cos(x+y)]

cos x cos y = 1/2[cos(x–y) + cos(x+y)]

sin x cos y = 1/2[sin(x+y) + sin(x−y)]

cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]

sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]

cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]

cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Sum or Difference of angles

cos (A + B) = cos A cos B – sin A sin B

cos (A – B) = cos A cos B + sin A sin B

sin (A+B) = sin A cos B + cos A sin B

sin (A -B) = sin A cos B – cos A sin B

tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]

tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]

cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]

cos(A+B) cos(A–B)=cos^2A–sin^2B=cos^2B–sin^2A

sin(A+B) sin(A–B) = sin^2A–sin^2B=cos^2B–cos^2A

Multiple and Submultiple angles

sin2A = 2sinA cosA = [2tan A /(1+tan²A)]

cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]

tan 2A = (2 tan A)/(1-tan²A)

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