Question

$\frac{ \cos \: a}{1 + \sin \: a} + \: \frac{1 + \sin \: a }{ \cos \: a } \: = 2 \sec \: a$
$prove$
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Answer 1

Answer:

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Answer 2

Qᴜᴇsᴛɪᴏɴ :-

Prove :- cosA/(1+sinA) +(1+sinA)/cosA = 2secA

Sᴏʟᴜᴛɪᴏɴ :-

Taking LHS,

→ cosA/(1+sinA) +(1+sinA)/cosA

Adding By taking LCM of Denominator, we get,

→ {cosA*cosA + (1+sinA)*(1+sinA)} / {cosA(1+sinA)}

→ {cos²A + (1+sinA)²} / {cosA(1+sinA)}

Now, Using (a + b)² = a² + b² + 2ab in Numerator,

→ {cos²A + 1 + sin²A + 2sinA} / {cosA(1+sinA)}

→ {1+(sin²A + cos²A) + 2sinA} / {cosA(1+sinA)}

putting (sin²A + cos²A) = 1 Now,

→ { 1 + 1 + 2sinA} / {cosA(1+sinA)}

→ {2 + 2sinA} / {cosA(1+sinA)}

Taking 2 common from Numerator,

→ 2{ 1 + sinA } / {cosA(1+sinA)}

cancel (1 + sinA) from N & D,

→ 2/cosA

using (1/cosA) = secA in Last,

→ 2secA = RHS (Proved).