Question

cosec A (1 + cos A) (cosec A-cot A) = 1​

Answer 1

$\boxed{ \huge{\mathfrak{solution}}}$

To prove :

$\boxed{ \purple{\bold{\csc \alpha (1 + \cos \alpha)( \csc \alpha - \cot \alpha) = 1}}}$

LHS:

$\leadsto \csc \alpha (1 + \cos( \alpha ) )( \frac{1}{ \sin( \alpha ) } - \frac{ \cos( \alpha ) }{ \sin( \alpha ) } ) \\ \\ \leadsto \: \csc( \alpha ) (1 + \cos( \alpha )) ( \frac{1 - \cos( \alpha ) }{ \sin( \alpha ) }) \\ \\ \leadsto \: \frac{1}{ \sin( \alpha ) } (1 + \cos( \alpha ) )( \frac{1 - \cos( \alpha ) }{ \sin( \alpha ) } ) \\ \\ \leadsto \: \frac{1 - { \cos}^{2} (\alpha )}{ { \sin}^{2} (\alpha) } \\ \\ \leadsto \: \frac{ { \sin}^{2} \alpha }{ { \sin}^{2} \alpha } \\ \\ \leadsto \: \cancel{ \frac{ { \sin}^{2} \alpha}{ { \sin}^{2} \alpha }} 1 \\ \\ \leadsto \: 1$

LHS=RHS

$\huge\boxed{ \green{\boxed{ \bold{proved}}}}$

Answer 2

${\large\bf{\mid{\overline{\underline{To\:Prove:-}}}\mid}}$

• cosecA(1 + cosA)(cosecA-cotA = 1

$\Large\bold\star\underline{\underline\textbf{Formula\:used}}$

• Cosec A = 1/sin A
• cot A = Cos A / sin A
• (1 - Cos²A) = sin²A
• (a+b)(a-b) = (a² - b²)

$\Large\underline{\underline{\sf{Solution}:}}$

$\red{using \: all \: above \: formula \: in} \: \\ \green{LHS} \: we \: get = \: \:$

$cosecA(1 + cosA)(cosecA-cotA) \\ \\ \red{\boxed\implies} \: \frac{1}{ \sin A}(1 + cosA) \: ( \frac{1}{\sin A} - \frac{ \cos A }{ \sin A } ) \\ \\ \red{\boxed\implies} \: \frac{1}{ \sin A}(1 + cosA) \:( \frac{1 - \cos A}{\sin A} ) \\ \\ \red{\boxed\implies} \: \frac{(1 - {cos}^{2}A )}{ {sin}^{2} A } \\ \\ \red{\boxed\implies} \: \: \: \: \frac{ \cancel{{sin}^{2} A}}{ \cancel{{sin}^{2} A}} \\ \\ \red{\boxed\implies} \: \: \pink{\large\boxed{\boxed{\bold{1 = RHS }}}} \:$

$\huge\boxed{\bold{Hence, Proved}}$