Question

➯ Answer this question!​

Answer 1

$\large\underline{\sf{Solution-}}$

We know that,

In triangle ABC,

Sum of all interior angles of a triangle is supplementary.

$\rm :\longmapsto\:A + B + C = \pi$

Consider,

$\rm :\longmapsto\:cos\bigg(\dfrac{A + 2B + 3C}{2} \bigg) + cos\bigg(\dfrac{A - C}{2} \bigg)$

We know,

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

Using this identity,

We get

$\rm \: = \:2cos\bigg(\dfrac{\dfrac{A + 2B + 3C}{2} + \dfrac{A - C}{2} }{2} \bigg)\:cos\bigg(\dfrac{\dfrac{A + 2B + 3C}{2} - \dfrac{A - C}{2} }{2} \bigg)$

$\rm \: =\:2cos\bigg(\dfrac{\dfrac{A + 2B + 3C + A - C}{2}}{2} \bigg)\:cos\bigg(\dfrac{\dfrac{A + 2B + 3C - A + C}{2}}{2} \bigg)$

$\rm \: = \: \: \:2cos\bigg(\dfrac{2A + 2B + 2C}{4} \bigg)cos\bigg(\dfrac{2B + 4C}{4} \bigg)$

$\rm \: = \: \: \:2cos\bigg(\dfrac{A + B + C}{2} \bigg)cos\bigg(\dfrac{B + 2C}{2} \bigg)$

$\rm \: = \: \: \:2cos\bigg(\dfrac{\pi}{2} \bigg)cos\bigg(\dfrac{B + 2C}{2} \bigg)$

$\rm \: = \: \: \:2 \times 0 \times cos\bigg(\dfrac{B + 2C}{2} \bigg)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: {\bigg \{ \because \:cos\dfrac{\pi}{2} = 0 \bigg \}}$

$\rm \: = \: \: \:0$

Hence,

$\bf :\longmapsto\:cos\bigg(\dfrac{A + 2B + 3C}{2} \bigg) + cos\bigg(\dfrac{A - C}{2} \bigg) = 0$

Thus,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf \: Option \: (a) \: is \: correct}}$

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)