Question

If $\sf \alpha+\beta+\gamma=\pi$
Then Prove the given statement.

$\sf cot\dfrac{\alpha}{2}+cot\dfrac{\beta}{2}+cot\dfrac{\gamma}{2}=cot\dfrac{\alpha}{2}\times cot\dfrac{\beta}{2}\times cot\dfrac{\gamma}{2}$
​

Answer 1

Basic Identities Used :-

$\boxed{ \sf \:cot\bigg(\dfrac{\pi}{2} - x \bigg) = tanx}$

$\boxed{ \sf \:cot(x + y) = \dfrac{cotx \: coty - 1}{coty + cotx}}$

Let's solve the problem now!!!

$\green{\large\underline{\bf{Solution-}}}$

Given that,

$\rm :\longmapsto\: \alpha + \beta + \gamma = \pi$

$\rm :\longmapsto\: \alpha + \beta = \pi - \gamma$

On dividing by 2, we get

$\rm :\longmapsto\:\dfrac{ \alpha }{2} + \dfrac{ \beta }{2} = \dfrac{\pi}{2} - \dfrac{ \gamma }{2}$

On applying cot on both sides, we get

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

$\rm :\longmapsto\:\dfrac{cot\bigg(\dfrac{ \alpha }{2} \bigg)cot\bigg(\dfrac{ \beta }{2}\bigg) - 1}{cot\bigg(\dfrac{ \beta }{2} \bigg) + cot\bigg(\dfrac{ \alpha }{2} \bigg)} = tan\bigg(\dfrac{ \gamma }{2}\bigg)$

$\rm :\longmapsto\:\dfrac{cot\bigg(\dfrac{ \alpha }{2} \bigg)cot\bigg(\dfrac{ \beta }{2}\bigg) - 1}{cot\bigg(\dfrac{ \beta }{2} \bigg) + cot\bigg(\dfrac{ \alpha }{2} \bigg)} = \dfrac{1}{cot\bigg(\dfrac{ \gamma }{2}\bigg)}$

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

$\rm\:cot\bigg(\dfrac{\alpha}{2}\bigg)cot\bigg(\dfrac{\beta}{2} \bigg)cot\bigg(\dfrac{\gamma}{2}\bigg) = cot\bigg(\dfrac{\gamma}{2}\bigg) + cot\bigg(\dfrac{\beta}{2} \bigg) + cot\bigg(\dfrac{\alpha}{2} \bigg)$

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

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)