Question

14) If A is not an integral
multiple of pie by 2 prove that
tanA+cot A = 2 cosecA
ii) cot A-tana - 2 cot 2 A.​

Answer 1

Basic Identities Used :-

$\boxed{ \sf \:cotx = \dfrac{1}{tanx}}$

$\boxed{ \sf \:sin2x = \dfrac{2tanx}{1 + {tan}^{2}x}}$

$\boxed{ \sf \:tan2x = \dfrac{2tanx}{1 - {tan}^{2}x}}$

$\boxed{ \sf \:cosecx = \dfrac{1}{sinx}}$

Let's solve the problem now!!!

$\large\underline{\bold{Given \:Question - 1}}$

Prove that

$\red{\rm :\longmapsto\:tanA + cotA = 2cosec2A}$

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

Consider,

$\rm :\longmapsto\:tanA + cotA$

$\rm \: = \: \: tanA + \dfrac{1}{tanA}$

$\rm \: = \: \: \dfrac{ {tan}^{2}A + 1}{tanA}$

$\rm \: = \: \: 2 \times \dfrac{ {tan}^{2}A + 1}{2tanA}$

$\rm \: = \: \: 2 \times \dfrac{1}{sin2A}$

$\rm \: = \: \: 2 \: cosec2A$

Hence Proved!!

$\large\underline{\bold{Given \:Question - 2}}$

Prove that

$\green{\bf :\longmapsto\:cotA - tanA = 2 \: cot2A}$

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

$\rm :\longmapsto\:cotA - tanA$

$\rm \: = \:\:\dfrac{1}{tanA} - tanA$

$\rm \: = \:\:\dfrac{1 - {tan}^{2}A }{tanA}$

$\rm \: = \:\:\dfrac{1 - {tan}^{2}A }{2tanA} \times 2$

$\rm \: = \: \: 2 \times \dfrac{1}{tan2A}$

$\rm \: = \: \: 2 \: cot2A$

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)