Question

A
Prove that cosA/(1 + sin A)+(1 + sin A)/cosA= 2 sec A​

Answer 1

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★ Sᴏʟᴜᴛɪᴏɴ :-

We need to prove,

[ cosA/(1 + sinA) ] + [ ( 1 + sinA )/cosA ] = 2 secA

Simplifying we have,

→ [ ( 1 + sinA )² + cos²A ] /( 1 + sinA ) cosA

→ [ 1² + sin²A + 2sinA + cos²A ] / cosA + sinAcosA

[°.° sin²A + cos²A = 1 ]

→ [ 1 + 1 + 2sinA ] / ( 1 + sinA )cosA

→ [ 2 + 2sinA ] / ( 1 + sinA )cosA

→ 2[ 1 + sinA ] / ( 1 + sinA )cosA

→ 2/cosA

It can be written as

→ 2 × [ 1/cosA ]

[°.° 1/cosA = secA ]

→ 2secA

Now , comparing with RHS

2secA = 2secA

LHS = RHS

Hence , proved !

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Answer 2

$\huge \underline{ \underline{ \bf \purple{Solution :}}}$

$\longrightarrow \: \sf\frac{ \cos}{1 + \sin} + \frac{1 + \sin}{ \cos} \\ \\ \longrightarrow \: \sf \frac{ { (\cos)}^{2} + {(1 + \sin)}^{2} }{(1 + \sin)( \cos)} \\ \\\longrightarrow \: \sf \frac{ {\cos}^{2} + {1 + 2 \sin + \sin}^{2} }{(1 + \sin)( \cos)} \\ \\\longrightarrow \: \sf \frac{ 1+ 1 + 2\sin}{(1 + \sin)( \cos)} \\ \\ \longrightarrow \: \sf\frac{ 2+ 2\sin}{(1 + \sin)( \cos)} \\ \\ \longrightarrow \: \sf \frac{2 \:( \cancel{ 1 + \sin})}{ ( \cancel{1 + \sin})\cos} \\ \\\longrightarrow \: \sf \frac{2}{ \cos} \\ \\ \longrightarrow \: \sf2 \sec$

$\LARGE \underline{ \underline{ \bf \orange {Hence \: Proved}}}$