Question

Prove that
sin A / 1 + cos A
= cosec A - cot A​

Answer 1

${\large {\underline {\blue {\bf {Question}}}}}$

Prove that

$\bf \: \dfrac{sinA}{1 + cosA} = cosecA - cotA$

${\huge {\underline {\blue {\bf {Answer}}}}}$

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

$(1) . \: \boxed{ \pink{\bf \: {sin}^{2} x + {cos}^{2}x = 1 }}$

$(2) . \: \boxed{ \purple{\bf \: cosecx \: = \: \dfrac{1}{sinx} }}$

$(3). \: \: \boxed{ \red{\bf \: secx \: = \: \dfrac{1}{cosx} }}$

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

Consider

$\bf \longmapsto \: \bf \:LHS$

$\bf \longmapsto \: \bf \:\dfrac{sinA}{1 + cosA}$

$\rm :\implies\:\dfrac{sinA}{1 + cosA} \times \dfrac{1 - cosA}{1 - cosA}$

$\rm :\implies\:\dfrac{sinA(1 - cosA)}{1 - {cos}^{2} A}$

$\rm :\implies\:\dfrac{sinA(1 - cosA)}{ {sin}^{2} A}$

$\rm :\implies\:\dfrac{1 - cosA}{sinA}$

$\rm :\implies\:\dfrac{1}{sinA} - \dfrac{cosA}{sinA}$

$\bf\implies \:cosecA - cotA$

$\bf \longmapsto \: \bf \:RHS$

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

Additional Information:-

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

Answer 2

Answer:

Your Answer Is Given Below

Step-by-step explanation:

LHS= sin A / 1 + cos A

= (sin A / 1 + cos A) × (1 - cos A / 1 - cos A)------{Rationalising the denominator}

= sin A (1 - cos A) / 1 - cos^2 A

= sin A (1 - cos A) / sin^2 A----{1st identity}

= 1 / sin A - cos A / sin A

= cosec A - cot A-----{Since 1 / sin A= cos A, cos A / sin A= cot A}

= RHS

THEREFORE, sin A / 1 + cos A= cosec A - cot A

HENCE PROVED.........