Question

1.(cosec²A-1) tan²A =1 prove that​

Answer 1

$\huge \mathfrak \color{red}{ Question:-}$

Prove the following :-

$\tt ( {cosec}^{2} A-1)tan^2=1$

$\huge \mathfrak \color{red}{Solution:-}$

$\: \: \: \: \: \: \: \: {\color{blue}\implies}\tt LHS \: : \: (cosec^2A-1)tan^2A=1$

$\: \: \: \: \: \: \: {\color{blue}\implies}\tt (1 + cot^2A - 1)tan^2A=1 \: {\color{blue}[\because \: cosec^2A=1+cot^2A]}$

$\: \: \: \: \: \: \: \: {\color{blue}\implies}\tt (cot^2A)(tan^2A)=1$

$\: \: \: \: \: \: \: \: {\color{blue}\implies}\tt ( \dfrac{1}{tan^2A} )(tan^2A)=1 \: {\color{blue}[\because \: tan^2A= \dfrac{1}{cot^2A} }$

$~~~~~~~~{\color{blue}\implies}\tt1 : RHS$

$\: \: \: \: ~ \: \: \: \: \: \underline{\boxed{\color{blue}\tt {\therefore \: LHS=RHS,~hence~proved!}}}$

$\huge \mathfrak {\color{red} Additional \: info:-}$

$\: \: \: \: \: \: \: \: {\color{blue}\implies}\tt cos( \dfrac{\pi}{2} - x ) = sinx$

$\: \: \: \: \: \: \: \: {\color{blue}\implies}\tt cosec (x\pm2\pi) = cosecx$

$\: \: \: \: \: \: \: \: {\color{blue}\implies}\tt tan( \dfrac{x}{2}) = \dfrac{(1 - cosx)}{sinx}$

$\: \: \: \: \: \: \: \: {\color{blue}\implies}\tt tan( \dfrac{\pi}{2} - x) = cotx$

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