Question

PROVE THAT :
(cot^2 A / cosec A-1) - 1 = cosec A
ANYONE PLZ HELP ​

Answer 1

Given :-

$\sf\ \dfrac{cot^2A}{cosecA-1}-1= CosecA$

To prove

• We have to prove the given equation :-

Solution

Taking LHS

$:\implies\it\ \dfrac{cot^2A}{cosecA-1}-1= CosecA\\ \\ \\ :\implies\sf\ \dfrac{cot^2A-(CosecA-1)}{CosecA-1}\\ \\ \\ :\implies\sf\ \ \dfrac{Cosec^2A-1-(CosecA-1)}{CosecA-1}\ \ \ \ \ \ \big[\because\ 1+cot^2A=cosec^2A\big]\\ \\ \\ :\implies\sf\ \dfrac{(CosecA-1)(CosecA+1)-(CosecA-1)}{CosecA-1}\\ \\ \\ :\implies\sf\ \dfrac{\cancel{CosecA-1}(CosecA\cancel{+1}\cancel{-1})}{\cancel{CosecA-1}}\\ \\ \\ :\implies\sf\ CosecA=RHS\ \ \ \ (Hence\ proved !!)$

Answer 2

To prove:-

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

─━─━─━─━─━─━─━─━─━─━─━─━─

$\begin{gathered}\Large{\bold{\purple{\underline{Formula \: Used \::}}}} \end{gathered}$

$1. \boxed{ \blue{ \bf \: {cot}^{2} = {cosec }^{2} A - 1}}$

$2. \boxed{ \red{ \bf \: {x}^{2} - {y}^{2} = (x + y)(x - y)}}$

─━─━─━─━─━─━─━─━─━─━─━─━─

$\large\underline\purple{\bold{Solution :- }}$

Consider LHS

$\tt \: \dfrac{ {cot}^{2}A }{cosec A - 1} - 1$

$\tt \:  = \dfrac{ {cosec }^{2} A - 1}{cosec A - 1} - 1$

$\tt \:  = \dfrac{(cosec A + 1) \: \: \cancel{(cosec A - 1)}}{ \cancel{cosec A - 1}} - 1$

$\tt \: =   cosec A \: + \: \cancel 1 \: - \: \cancel1$

$\tt \:  = cosec A$

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

─━─━─━─━─━─━─━─━─━─━─━─━─

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1