Question

if sin x + sin y = 4/5 and sin x - sin y = 2/7 prove that 5 tan (y + x)/2 + 14 tan( y - x)/2 = 0​

Answer 1

Answer:

= 0

Step-by-step explanation:

if sin x +

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ye dhfgv ISBN ugh ugh us chha

hence proved =00000000

Answer 2

Use of Identities :-

$\boxed{ \sf \: sinx + siny = 2sin\bigg(\dfrac{x + y}{2} \bigg)cos\bigg(\dfrac{x - y}{2} \bigg)}$

$\boxed{ \sf \: sinx - siny = 2cos\bigg(\dfrac{x + y}{2} \bigg)sin\bigg(\dfrac{x - y}{2} \bigg)}$

$\boxed{ \sf \: tan( - x) = - tanx}$

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

Given that,

$\rm :\longmapsto\:sinx + siny = \dfrac{4}{5} - - - (1)$

and

$\rm :\longmapsto\:sinx - siny = \dfrac{2}{7} - - - (2)$

On dividing equation (1) by equation (2), we get

$\rm :\longmapsto\:\dfrac{sinx + siny}{sinx - siny} = \dfrac{4}{5} \times \dfrac{7}{2}$

$\rm :\longmapsto\:\dfrac{2sin\bigg(\dfrac{x + y}{2} \bigg)cos\bigg(\dfrac{x - y}{2} \bigg)}{2cos\bigg(\dfrac{x + y}{2}\bigg) sin\bigg(\dfrac{x - y}{2} \bigg)} = \dfrac{14}{5}$

$\rm :\longmapsto\:\dfrac{sin\bigg(\dfrac{x + y}{2} \bigg)cos\bigg(\dfrac{x - y}{2} \bigg)}{cos\bigg(\dfrac{x + y}{2}\bigg) sin\bigg(\dfrac{x - y}{2} \bigg)} = \dfrac{14}{5}$

$\rm :\longmapsto\:\dfrac{tan\bigg(\dfrac{x + y}{2} \bigg)}{tan\bigg(\dfrac{x - y}{2} \bigg)} = \dfrac{14}{5}$

$\rm :\longmapsto\:5tan\bigg(\dfrac{x + y}{2} \bigg) = 14tan\bigg(\dfrac{x - y}{2} \bigg)$

$\rm :\longmapsto\:5tan\bigg(\dfrac{x + y}{2} \bigg) = - 14tan\bigg(\dfrac{y - x}{2} \bigg)$

$\rm :\longmapsto\:5tan\bigg(\dfrac{x + y}{2} \bigg) + 14tan\bigg(\dfrac{y - x}{2} \bigg) = 0$

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

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)