Question

please answer
I'll mark you as brainliest​

Answer 1

$\red{\underline{\underline{Answer:}}}$

$\sf{(secA+tanA)(1-sinA)=cosA}$

$\sf\pink{To \ find:}$

$\sf{(secA+tanA)(1-sinA)}$

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

$\sf{\implies{(secA+tanA)(1-sinA)}}$

$\sf{sec=\frac{1}{cos}}$

$\sf{tan=\frac{sin}{cos}}$

$\sf{Trignometric \ ratio}$

$\sf{\implies{(\frac{1}{cosA}+\frac{sinA}{cosA})(1-sinA)}}$

$\sf{\implies{(\frac{1+sinA}{cosA})(1-sinA)}}$

$\sf{\implies{\frac{1-sin^{2}A}{cosA}}}$

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

$\sf{... Trignometric \ identity}$

$\sf{\implies{\frac{cos^{2}A}{cosA}}}$

$\sf{\implies{cosA}}$

$\sf\purple{\tt{\therefore{(secA+tanA)(1-sinA)=cosA}}}$

Answer 2

$\huge\underline\mathfrak\red{Question:}$

$( \sec(A) + \tan(A) )(1 - \sin(A) ) = ?$

$A) \: \sec(A)$

$B) \: \sin(A)$

$C) \: \cosec(A)$

$D) \: \cos(A)$

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$\huge\underline\mathfrak\green{Answer:}$

$\rm\purple{D) \: \cos(A)}$

___________________________________________

$\sf\large\underline\pink{Given:}$

$( \sec(A) + \tan(A) )(1 - \sin(A) )$

___________________________________________

$\sf\large\underline\pink{To \ find:}$

$( \sec(A) + \tan(A) )(1 - \sin(A) ) = ?$

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$\huge\underline\mathfrak\blue{Solution:}$

$\rm{given }$

⇒$( \sec(A) + \tan(A) )(1 - \sin(A) )$

$\rm{We \ know ,}$

$\sec( \alpha ) = \frac{1}{ \cos( \alpha ) }$

$\rm{and}$

$\tan( \alpha ) = \frac{ \sin( \alpha ) }{ \cos( \alpha ) }$

$\rm{By \ using \ trigonometric \ formulae }$

⇒$( \frac{1}{ \cos(A) } = \frac{ \sin(A) }{ \cos(A) } )(1 - \sin(A) )$

⇒$( \frac{1 + \sin(A) }{ \cos(A) } )(1 - \sin(A) )$

⇒$\frac{1 - { \sin(A) }^{2} }{ \cos(A) }$

$but \: \: 1 - { \sin(A) }^{2} = { \cos(A) }^{2}$

⇒$\frac{ { \cos(A) }^{2} }{ \cos(A) }$

⇒$\cos(A)$

$\rm\orange{( \sec(A) + \tan(A) )(1 - \sin(A) ) = \cos(A)}$

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$\rm\red{Hope \ it \ helps \ :))}$