Question

Prove that : cosec A (1+ cos A)(cosec A- cot A) = 1
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Answer 1

$\huge\sf\red{Question\::}$

$\sf Prove\: that\: :$

$\sf{cosecA (1+cosA)(cosecA-cotA)=1}$

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$\huge\sf\purple{Solution\: :}$

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$\sf L.H.S\: :\:cosecA (1+cosA)(cosecA-cotA)$

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$\longrightarrow\:\:\:\sf{\dfrac{1}{sinA}(1+cosA)\Bigg(\dfrac{1}{sinA}-\dfrac{cosA}{sinA}\Bigg)}$

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$\longrightarrow\:\:\:\sf{\dfrac{1}{sinA}(1+cosA)\Bigg(\dfrac{1-cosA}{sinA}\Bigg)}$

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$\longrightarrow\:\:\:\sf{\dfrac{(1+cosA)(1-cosA)}{sin^2A}}$

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${\boxed{\sf{(a+b)(a-b)= a^2 - b^2}}}$

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$\longrightarrow\:\:\:\sf{\dfrac{1-cos^2A}{sin^2A}}$

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${\boxed{\sf{sin^2 A = 1 - cos^2 A}}}$

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$\longrightarrow\:\:\:\sf{\dfrac{sin^2A}{sin^2A}}$

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$\longrightarrow\:\:\:\sf{1}$

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$\sf\pink{ L.H.S\: =\: R.H.S\: (Hence\: Proved)}$

Answer 2

Let, A = α

$⟹ \csc\alpha (1 + \cos \alpha ) ( \csc\alpha - \cot \alpha )$

• [cosec α = 1/sin α ]
• [cot α = cos α/sin α ]

$⟹ \frac{1}{ \sin \alpha } (1 + \cos \alpha ) ( \frac{1}{ \sin \alpha } - \frac{ \cos \alpha }{ \sin \alpha } )$

$⟹( \frac{1 + \cos \alpha }{ \sin \alpha } )( \frac{1 - \cos \alpha }{ \sin \alpha } )$

• (a + b)(a - b) = a² - b²

$⟹ \frac{ {(1 - \cos }^{2} \alpha )} {{ \sin }^{2} \alpha }$

• 1 - cos² α = sin² α

$⟹ \frac{ { \sin }^{2} \alpha }{ { \sin }^{2} \alpha }$

$⟹1$

Step-by-step explanation:

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