Question

$\sf \: \: \dfrac{ \tan(a) }{ \sec(a) - 1 } + \dfrac{ \tan(a) }{ \sec(a) + 1} = 2 \cosec(a)$


Need prove :(​

Answer 1

${\purple{\boxed{\large{\bold{Formula's}}}}}$

$\\ \sf • \: \tan(a) = \frac{ \sin(a) }{ \cos(a) } \\ \\ \sf • \: \sec(a) = \frac{1}{ \cos(a) } \\ \\ \sf • \: \cosec(a) = \frac{1}{ \sin(a) } \\ \\ \sf •\: 1 - \cos {}^{2} (a) = \sin {}^{2} (a)$

$\underline{ \mathfrak{ \: To\:Prove:- \: }}$

$\sf \dashrightarrow \: \: \dfrac{ \tan(a) }{ \sec(a) - 1 } + \dfrac{ \tan(a) }{ \sec(a) + 1} = 2 \cosec(a)$

$\underline{ \mathfrak{ \: Proof :- \: }}$

$\sf \dashrightarrow \: \: \dfrac{ \tan(a) }{ \sec(a) - 1 } + \dfrac{ \tan(a) }{ \sec(a) + 1}$

$\sf \dashrightarrow \: \frac{ \dfrac{ \sin(a) }{ \cos(a) } }{ \dfrac{1}{ \cos(a) } - 1 } \: \: + \: \: \frac{ \dfrac{ \sin(a) }{ \cos(a) } }{ \dfrac{1}{ \cos(a) } + 1}$

$\sf \dashrightarrow \: \dfrac{ \dfrac{ \sin(a) }{ \cos(a) } }{ \dfrac{1 - \cos(a) }{ \cos(a) } } \: \: + \: \: \: \dfrac{ \dfrac{ \sin(a) }{ \cos(a) } }{ \dfrac{1 + \cos(a) }{ \cos(a) } }$

$\sf \dashrightarrow \: \dfrac{ \sin(a) }{ \cancel{\cos(a)} } \times \dfrac{ \cancel{\cos(a)} }{1 - \cos(a) } \: \: + \: \: \dfrac{ \sin(a) }{ \cancel{\cos(a)} } \times \dfrac{ \cancel{\cos(a)} }{1 + \cos(a) }$

$\sf \dashrightarrow \: \dfrac{ \sin(a) }{1 - \cos(a) } \: \: + \: \: \: \dfrac{ \sin(a) }{1 + \cos(a) }$

$\sf\dashrightarrow \: \dfrac{ \sin(a) (1 + \cos(a) + \sin(a) (1 - \cos(a)}{(1 - \cos(a)(1 + \cos(a) }$

$\sf \dashrightarrow \: \dfrac{ \sin(a) (1 + \cos(a) + 1 - \cos(a )}{(1 - \cos {}^{2} (a )}$

$\sf \dashrightarrow \: \dfrac{ 2\sin(a) }{ \sin {}^{2} (a) } = \dfrac{2}{ \sin(a) } = 2 \cosec(a)$

ㅤ$\;\;\underline{\textbf{\textsf{Hence -}}}$

$\leadsto \ {\boxed{\tt{LHS = RHS}}}$

$\therefore{ \underline{\bf{(Proved.....!!)}}}$

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Answer 2

$\frac{tan \: a \: (sec \: a \: + 1) \: + \: tan \: a \: (sec \: a \: - 1)}{(sec \: a \: - \: 1) \: (sec \: a \: + \: 1)} \\ \\ = > \frac{tan \: a \: (sec \: a \: + \: 1 \: + \: sec \: a \: - \: 1)}{ {sec}^{2} \: - \: 1} \\ \\ = > \frac{tan \: a \: (2 \: sec \: a)}{ {sec}^{2} \: - \: 1} \\ \\ = > \frac{2 \: sec \: a \: \times \: tan \: a }{1 \: + \: {tan}^{2} \: - \: 1 } \\ \\ = > \frac{2 \: sec \: a \: \: tan \: a}{ {tan}^{2} \: a } \\ \\ = > \frac{2 \: sec \: a}{tan \: a } \\ \\ = > 2 \: \: sec \: a \: \: cot \: a \\ \\ = > 2 \: \times \: \frac{1}{cos \: a} \times \frac{cos \: a}{sin \: a} \\ \\ = > 2 \: \times \: cosec \: a \:$

Done