Math, asked by sudhasudhakar10, 9 months ago

SinAtanA/1-cosA=1+secA

Answers

Answered by Anonymous
15

{\underline{\underline{\large{\mathtt{QUESTION:-}}}}}

Prove that,

\sf\frac{sinA\:tanA}{1-cosA}=1+secA

{\underline{\underline{\large{\mathtt{SOLUTION:-}}}}}

Taking L.H.S ,

\sf\frac{sinA\:tanA}{1-cosA}

★we know that tanA = sinA/cosA★

\implies\sf\frac{sinA\:\frac{sinA}{cosA}}{1-cosA}

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

★We know that 1-cos²A = sin²A★

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

★Apply a²-b²=(a+b)(a-b)★

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

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

\implies\sf\frac{1+cosA}{cosA}

★Apply the rule of separation★

\implies\sf\frac{1}{cosA}+\frac{cosA}{cosA}

\implies\sf{secA+1}

\implies\sf{1+secA\:(Proved)}

L.H.S = R.H.S

Answered by MrBhukkad
4

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 \tt{ \underline{ \underline{ \red{Question}}}}: -  \\  \bf{Prove \: that,} \\  \bf{ \frac{sinAtanA}{1 - cosA}  = 1 + secA} \\  \tt{ \underline{ \underline{ \orange{Answer}}}}:  -

Taking L.H.S.,

 \\  \\  \bf{</u></strong><strong><u>Formulas</u></strong><strong><u> \: used} \\  \boxed{ \bf{ 1 -  {cos}^{2}</u></strong><strong><u>A</u></strong><strong><u> =  {sin}^{2}</u></strong><strong><u>A</u></strong><strong><u>}} \\  \boxed{ \bf{ {a}^{2}  -  {b}^{2} = (a + b)(a - b) +   }} \\  \\  \longrightarrow \bf {\frac{sin</u></strong><strong><u>A</u></strong><strong><u>tan</u></strong><strong><u>A</u></strong><strong><u>}{1 - cos</u></strong><strong><u>A</u></strong><strong><u>} } \\  \longrightarrow \bf{ \frac{sin</u></strong><strong><u>A</u></strong><strong><u> \frac{sin</u></strong><strong><u>A</u></strong><strong><u>}{cos</u></strong><strong><u>A</u></strong><strong><u>} }{1 - cos</u></strong><strong><u>A</u></strong><strong><u>} } \\  \longrightarrow \bf{ \frac{ \frac{ {sin}^{2}</u></strong><strong><u>A</u></strong><strong><u> }{cos</u></strong><strong><u>A</u></strong><strong><u>} }{1 - cos</u></strong><strong><u>A</u></strong><strong><u>} } \\  \longrightarrow \bf{ \frac{ \frac{1 -  {cos}^{2} </u></strong><strong><u>A</u></strong><strong><u>}{cos</u></strong><strong><u>A</u></strong><strong><u>} }{1 - cos</u></strong><strong><u>A</u></strong><strong><u>} } \\  \longrightarrow \bf{ \frac{ \frac{(1 + cos</u></strong><strong><u>A</u></strong><strong><u>)(1 - cos</u></strong><strong><u>A</u></strong><strong><u>)}{cos</u></strong><strong><u>A</u></strong><strong><u>} }{1 - cos</u></strong><strong><u>A</u></strong><strong><u>} } \\  \longrightarrow \bf{ \frac{(1 + cos</u></strong><strong><u>A</u></strong><strong><u>) \cancel{(1 - cos</u></strong><strong><u>A</u></strong><strong><u>)}}{cos</u></strong><strong><u>A</u></strong><strong><u> \cancel{(1 - cos</u></strong><strong><u>A</u></strong><strong><u>)}} } \\  \longrightarrow \bf{ \frac{1}{cos</u></strong><strong><u>A</u></strong><strong><u>}  +  \frac{cos</u></strong><strong><u>A</u></strong><strong><u>}{cos</u></strong><strong><u>A</u></strong><strong><u>} } \\  \longrightarrow \bf{sec</u></strong><strong><u>A</u></strong><strong><u> + 1} \\  \longrightarrow \bf{1 + sec</u></strong><strong><u>A</u></strong><strong><u>} \: (</u></strong><strong><u>Proved</u></strong><strong><u>)

L.HS = R.H.S.

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