Question

prove that 1+cos2A/1-cos2A=cot^2 A​

Answer 1

TO PROVE :-

$\\ \tt \: \dfrac{1 + cos2A}{1 - cos2A} = {cot}^{2}A$

IDENTITY USED :-

★ cos2A = cos²A - sin²A

★ 1 = sin²A + cos²A

★ cosA/sinA = cotA

SOLUTION :-

$\\ \tt \: L.H.S = \dfrac{1 + cos2A}{1 - cos2A} \\ \\ \\ \bigstar\boxed{ \sf \: cos2A = {cos}^{2}A - {sin}^{2}A } \\ \\ \\ \tt \implies \: \dfrac{1 + ( {cos}^{2}A - {sin}^{2}A) }{1 - {(cos}^{2}A - {sin}^{2}A )} \\ \\ \\ \bigstar \boxed{ \sf \: 1 = {sin}^{2}A + {cos}^{2}A } \\ \\ \\ \tt \implies \: \dfrac{ \cancel{{sin}^{2}A} + {cos}^{2}A + {cos}^{2}A - \cancel {{sin}^{2}A }}{ {sin}^{2}A + \cancel{{cos}^{2}A }- \cancel {{cos}^{2}A} + {sin}^{2}A } \\ \\ \\ \tt \implies \: \dfrac{2 {cos}^{2}A }{2 {sin}^{2}A } \\ \\ \\ \bigstar \boxed{ \sf \: \dfrac{cosA}{sinA} = cotA}$

$\\ \implies \tt \: {cot}^{2}A = R.H.S \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: (verified)$

MORE IDENTITIES :-

★ 1 + cot²A = cosec²A

★ 1 + tan²A = sec²A

★ tanA = sinA/cosA

★ sinA = 1/cosecA

★ cosA = 1/secA

★ sin2A = 2sinA.cosA

Answer 2

$\boxed{ \red{ \rm \: Prove \: that \: : \: \dfrac{1 + cos2A}{1 - cos2A} = {cot}^{2} A \: }}$

We know,

$\boxed{ \rm \: cos2A = {cos}^{2} A - {sin}^{2} A = 1 - {2sin}^{2} A = 2{cos}^{2}A - 1 }$

$\boxed{ \pink{ \rm : \implies \:1 - cos2A = {2sin}^{2} A}}$

$\boxed{ \pink{ \rm : \implies \:1 + cos2A = {2cos}^{2} A}}$

$\boxed{ \pink{ \rm \implies \: \dfrac{cosA}{sinA} = cotA \: }}$

Now,

Consider LHS, we have

$\rm : \implies \:\dfrac{1 + cos2A}{1 - cos2A}$

$\rm : \implies \:\dfrac{ \cancel2 \: {cos}^{2}A }{ \cancel2 \: {sin}^{2} A}$

$\rm : \implies \: {cot}^{2} A$

$\bf : \implies \: LHS = RHS$

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

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)