Question

prove that tan² theta/ 1+tan²theta + cot²theta/ 1+cot²theta =1​

Answer 1

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❄ Question :

$\rm{\bullet {Prove that, \dfrac{tan^2\theta}{1+tan^2\theta} +\dfrac{cot^2 \theta}{1+cot^2\theta}=1}}$

$\huge\red{\sf{\underline{\underline{Required \: Answer}}}}$

$\: \longrightarrow \displaystyle \sf \: \frac{ {tan}^{2} x}{1 + {tan}^{2}x } + \frac{ {cot}^{2}x }{1 + {cot}^{2} x}$

$\longrightarrow \displaystyle \sf \: \frac{ \frac{ {sin}^{2}x }{ {cos}^{2}x } }{1 + \frac{ {sin}^{2}x }{ {cos}^{2}x } } + \frac{ \frac{ {cos}^{2}x }{ {sin}^{2}x } }{1 + \frac{ {cos}^{2}x }{ {sin}^{2} x} }$

$\longrightarrow \displaystyle \sf \: \frac{ \frac{ {sin}^{2}x }{ {cos}^{2}x } }{ \frac{ {cos}^{2}x + {sin}^{2}x }{ {cos}^{2}x } } + \frac{ \frac{ {cos}^{2} x}{ {sin}^{2} x} }{ \frac{ {sin}^{2}x + {cos}^{2}x }{ {sin}^{2} x} }$

$\longrightarrow \displaystyle \sf \: \frac{ \frac{ {sin}^{2}x}{ \cancel{ {cos}^{2}x} } }{ \frac{ {cos}^{2} x + {sin}^{2} x}{ \cancel {cos}^{2}x } } + \frac{ \frac{ {cos}^{2}x }{ \cancel {sin}^{2}x} }{ \frac{ {sin}^{2}x + {cos}^{2}x }{ \cancel {sin}^{2} x} }$

$\\ \longrightarrow \displaystyle \sf \: \frac{ {sin}^{2}x }{ {cos}^{2}x + {sin}^{2}x } + \frac{ {cos}^{2}x }{ {sin}^{2}x + {cos}^{2}x }$

$\longrightarrow \displaystyle \sf \: \frac{ {sin}^{2}x + {cos}^{2} x}{ {sin}^{2}x + {cos}^{2}x }$

Properties :

$\bf{ \bullet{ {sin}^{2} x + {cos}^{2} x = 1}}$

$\bf{ \bullet{ \: {cos}^{2} x = 1 - {sin}^{2} x}}$

$\longrightarrow \displaystyle \sf \: \frac{ {sin}^{2}x + {cos}^{2}x }{ {sin}^{2}x + {cos}^{2}x } = \frac{1}{1} = 1$

$\longrightarrow \displaystyle {\ \boxed{ \bigstar \: \: \pink{ \underline{ \bf{the \: answer \: will \: be \: 1}}}}}$

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