Question

If A+B+C=180°then the value of TanA/2TanB/2+TanB/2 TanC/2+Tanc/2TanA/2 =?​

Answer 1

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

If A+B+C=180°, then the value of TanA/2TanB/2+TanB/2 TanC/2+Tanc/2TanA/2 =?

─━─━─━─━─━─━─━─━─━─━─━─━─

$\bf \:\huge \red{AηsωeR } ✍$

─━─━─━─━─━─━─━─━─━─━─━─━─

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

$\bf \:  \: A \: + \: B \: + \: C \: = \: 180°$

─━─━─━─━─━─━─━─━─━─━─━─━─

$\large\underline\blue{\bold{To \:  Find :-  }}$

$\sf \:the \: value \: of  tan\dfrac{A}{2}tan \dfrac{B}{2} + tan\dfrac{B}{2} tan\dfrac{C}{2} +tan \dfrac{C}{2}tan \dfrac{A}{2}$

─━─━─━─━─━─━─━─━─━─━─━─━─

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

$\sf \:  1. \: \boxed{\bf \:  ⟼ tan(x + y) = \dfrac{tanx + tany}{1 - tanx \: tany}}$

$\sf \:  2. \: \boxed{\bf \:  ⟼ tan(90° - x) = cotx}$

─━─━─━─━─━─━─━─━─━─━─━─━─

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

$\bf \:  Since, \: A \: + \: B \: + \: C \: = \: 180°$

$\sf \:  ⟼A \: + \: B \: = \: 180° \: - \: C$

$\bf\implies \:\dfrac{A}{2} + \dfrac{B}{2} + \dfrac{C}{2} = 90°$

$\sf \:  ⟼\dfrac{A}{2} + \dfrac{B}{2} = 90° - \dfrac{C}{2}$

$\bf\implies \:tan(\dfrac{A}{2} + \dfrac{B}{2} ) = tan(90° - \dfrac{C}{2})$

$\sf \:  ⟼\dfrac{tan\dfrac{A}{2} + tan\dfrac{B}{2} }{1 - tan\dfrac{A}{2}tan\dfrac{B}{2}} = cot\dfrac{C}{2}$

$\sf \:  ⟼\dfrac{tan\dfrac{A}{2} + tan\dfrac{B}{2} }{1 - tan\dfrac{A}{2}tan\dfrac{B}{2}} = \: \dfrac{1}{tan\dfrac{C}{2}}$

$\sf \:  ⟼tan\dfrac{A}{2}tan\dfrac{C}{2} + tan\dfrac{B}{2}tan\dfrac{C}{2} = 1 - tan\dfrac{A}{2}tan\dfrac{B}{2}$

$\sf\implies \:tan\dfrac{A}{2}\:tan\dfrac{B}{2} + \:tan\dfrac{B}{2} \:tan\dfrac{C}{2} + \:tan\dfrac{C}{2}\:tan\dfrac{A}{2} = 1$

─━─━─━─━─━─━─━─━─━─━─━─━─

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)

─━─━─━─━─━─━─━─━─━─━─━─━─