Question

Kripya ise solve kre!​

Answer 1

Given Question :-

$\sf \: If \: cos\theta\:=\cos( \alpha ) \cos( \beta ),\: then\: tan\bigg(\dfrac{\theta+\alpha }{2} \bigg)tan\bigg(\dfrac{\theta-\alpha }{2}\bigg) =$

Solution

Identities Used :-

$\boxed{\red{\sf\:cosx + cosy = 2cos\bigg(\dfrac{x + y}{2} \bigg)cos\bigg(\dfrac{x - y}{2} \bigg)}}$

$\boxed{\red{\sf\:cosx - cosy = - 2sin\bigg(\dfrac{x + y}{2} \bigg)sin\bigg(\dfrac{x - y}{2} \bigg)}}$

$\boxed{\red{\sf\:1 - cosx = 2 {sin}^{2}\dfrac{x}{2}}}$

$\boxed{\red{\sf\:1 + cosx = 2 {cos}^{2}\dfrac{x}{2}}}$

Let's solve the problem now!!

Given that

$\rm :\longmapsto\:cos\theta = cos \alpha cos \beta$

$\rm :\longmapsto\:\dfrac{cos\theta}{cos \alpha } = cos \beta$

$\rm :\longmapsto\:\dfrac{cos\theta}{cos \alpha } = \dfrac{cos \beta }{1}$

Apply Componendo and Dividendo, we get

$\rm :\longmapsto\:\dfrac{cos\theta + cos \alpha }{cos\theta - cos \alpha } = \dfrac{cos \beta + 1}{cos \beta - 1}$

$\rm :\longmapsto\:\dfrac{2cos\bigg(\dfrac{\theta + \alpha }{2} \bigg)cos\bigg(\dfrac{\theta - \alpha }{2} \bigg)}{ - 2sin\bigg(\dfrac{\theta + \alpha }{2} \bigg)sin\bigg(\dfrac{\theta - \alpha }{2} \bigg)} = \dfrac{2 {cos}^{2}\dfrac{ \beta }{2} }{ - 2 {sin}^{2} \dfrac{ \beta }{2} }$

$\rm :\longmapsto\:\dfrac{cos\bigg(\dfrac{\theta + \alpha }{2} \bigg)cos\bigg(\dfrac{\theta - \alpha }{2} \bigg)}{ sin\bigg(\dfrac{\theta + \alpha }{2} \bigg)sin\bigg(\dfrac{\theta - \alpha }{2} \bigg)} = \dfrac{{cos}^{2}\dfrac{ \beta }{2} }{{sin}^{2} \dfrac{ \beta }{2} }$

$\rm :\longmapsto\:cot\bigg(\dfrac{\theta + \alpha }{2} \bigg)cot\bigg(\dfrac{\theta - \alpha }{2} \bigg) = {cot}^{2} \dfrac{ \beta }{2}$

$\rm :\implies\:tan\bigg(\dfrac{\theta + \alpha }{2} \bigg)tan\bigg(\dfrac{\theta - \alpha }{2} \bigg) = {tan}^{2} \dfrac{ \beta }{2}$

$\bf\implies \:Option \: (b) \: is \: correct$

Additional Information :-

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)