Question

cosec 2A + cosec 4A = cot A-cot 4A​

Answer 1

Question:

$\sf{Prove\:that\:cosec2A+cosec4=cotA-cot4A}$

Identities Used:

$\sf{cosecA=\dfrac{1}{sinA}}\\\sf{sin2A=2sinAcosA}\\\sf{\dfrac{cosA}{sinA}=cotA}\\\sf{cos2A=2cos^2A-1}$

Solution:

$\sf{Taking\:L.H.S.}$

$\sf{=cosec2A+cosec4A}$

$\sf{=\dfrac{1}{sin2A}+\dfrac{1}{sin4A}}$

$\sf{=\dfrac{1}{sin2A}+\dfrac{1}{2sin2Acos2A}}$

$\sf{Taking\:L.C.M.}$

$\sf{=\dfrac{2cos2A+1}{2sin2Acos2A}}$

$\sf{=\dfrac{2cos2A+1+cos4A-cos4A}{sin4A}}$

$\sf{=\dfrac{2cos2A+1+cos4A}{sin4A}-\dfrac{cos4A}{sin4A}}$

$\sf{=\dfrac{2cos2A+1+2cos^2A-1}{sin4A}-cot4A}$

$\sf{=\dfrac{2cos2A\left(1+cos2A\right)}{2sin2Acos2A}-cot4A}$

$\sf{=\dfrac{1+2cos^2A-1}{sin2A}-cot4A}$

$\sf{=\dfrac{2cos^2A}{2sinAcosA}-cot4A}$

$\sf{=\dfrac{cosA}{sinA}-cot4A}$

$\sf{=cotA-cot4A}$

$\bf{\sf{HENCE\:PROVED}}$

Answer 2

Step-by-step explanation:

Given: cosec 2A + cosec 4A = cot A-cot 4A

To prove: L.H.S = R.H.S

† Taking L.H.S

⇒Cosec 2 A + Cot 4 A

⇒Cosec 2 A + Cot 4 A⇒Cosec 2 A + ( Cot 2 A )2

⇒ Cosec 2A + (Cosec 2A–1)2

⠀⠀⠀(Since, 1 + cot theta= Cosec theta)

→ Cosec 2A +Cosec 4A + 1 –2Cosec 2A

→Cosec 4A + 1 – Cosec 2A

→Cosec 4A + 1 – (1+ Cot 2A)

→ Cosec 4A + 1 – 1 – Cot 2A

→ Cosec 4A – Cot 2A

Or,

Cosec 2A + Cosec 4A = Cot A – Cot 4A

Hence, L.H.S = R.H.S