Question

cos (n + 1) a cos (n-1) a + sin (n + 1)a sin (n - 1) a =
a) con2na
b) sin2n a
c) cos2a
d) sin2 a
please say the correct explanation. ​

Answer 1

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

Given that

$\rm :\longmapsto\:cos(n + 1)a \: cos(n - 1)a + sin(n + 1)a \: sin(n - 1)a$

Let assume that

$\rm :\longmapsto\:(n + 1)a = x \\ \rm :\longmapsto\:(n - 1)a = y$

So, given expression reduced to

$\rm :\longmapsto\:cosx \: cosy \: + \: sinx \: siny$

$\rm \: = \: \:cos(x - y)$

$\red{\bigg \{ \because \:cosx \: cosy + sinx \: siny \: = \: cos(x - y) \bigg \}}$

$\rm \: = \: \:cos\bigg((n + 1)a - (n - 1)a\bigg)$

$\rm \: = \: \:cos(na + a - na + a)$

$\rm \: = \: \:cos2a$

Hence,

$\rm :\longmapsto\:cos(n + 1)a \: cos(n - 1)a + sin(n + 1)a \: sin(n - 1)a$

$\rm \: = \: \:cos2a$

Hence,

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

$\blue{\boxed{ \bf{ \: Additional \: \: Information}}}$

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)