Question

cos A / 1 + sin A + 1 + sinA / cos A
= 2 sec A
Prove that L H S = R H S​

Answer 1

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

Prove that,

$\sf{\frac{cosA}{1+sinA}+\frac{1+sinA}{cosA}=\:2secA}$

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

Taking L.H.S ,

$\sf{\frac{cosA}{1+sinA}+\frac{1+sinA}{cosA}}$

$\implies\sf\frac{cos^2A+(1+sinA)^2}{cosA\:(1+sinA)}$

★Apply (a+b)² = a²+2ab+b²★

$\implies\sf\frac{cos^2A+1+2sinA+sin^2A}{cosA\:(1+sinA)}$

$\implies\sf\frac{cos^2A+sin^2A+2sinA+1}{cosA\:(1+sinA)}$

★We know sin²A +cos²A = 1★

$\implies\sf\frac{1+2sinA+1}{cosA\:(1+sinA)}$

$\implies\sf\frac{2+2sinA}{cosA\:(1+sinA)}$

$\implies\sf\frac{2(1+sinA)}{cosA\:(1+sinA)}$

$\implies\sf\frac{2}{cosA\:}$

$\implies\sf{2secA\:(Proved)}$

L.H.S = R.H.S ( Proved)

Answer 2

$\large\underline\mathrm{Solution}$

$\huge \mathfrak{L.H.S:-}$

$\implies$ cosA/1+ sinA + 1+ sinA/cosA

$\implies$ cos²A + (1+ sinA)²/cos A(1 + sinA)

$\large\underline\mathrm{Using \: \: formula.(a \: + \: b)² \: = \: a² \: + \: 2ab \: + \: b²}$

$\implies$ cos²A + 1 + 2sinA + sin²A/cosA (1 + sinA)

$\implies$ cos²A + sin²A + 2sinA + 1 / cosA (1 + sinA)

$\large\underline\mathrm{Now, \: [ \: sin²A \: + \: cos²A]}$

$\implies$ 1 + 2sinA + 1 / cosA(1 + sinA)

$\implies$ 2 + 2sinA / cosA(1 + sinA)

$\implies$ 2(1 + sinA)/cos(1 + sinA)

$\implies$ 2/cosA

$\implies$ 2secA

$\large\underline\mathrm{L.H.L \: = \: R.H.S}$

$\large\underline\mathrm{Hope \: it \: helps \: you \: plz \: mark \: me \: brainlist}$