Question

cos a/1+sin a + 1+sin a/cos a= 2 sec a​

Answer 1

Answer:

this your answer

Step-by-step explanation:

LHS = Cos A/(1+ sin A) + (1+ sin A)/cos A  

= begin mathsize 16px style fraction numerator cosA over denominator 1 plus sinA end fraction plus fraction numerator 1 plus sinA over denominator cosA end fraction end style

= begin mathsize 16px style fraction numerator cos squared straight A plus open parentheses 1 plus sinA close parentheses squared over denominator cosA open parentheses 1 plus sinA close parentheses end fraction end style

= begin mathsize 16px style fraction numerator cos squared straight A plus 1 plus 2 sinA plus sin squared straight A over denominator cosA open parentheses 1 plus sinA close parentheses end fraction end style

= begin mathsize 16px style fraction numerator 1 plus 1 plus 2 sinA over denominator cosA open parentheses 1 plus sinA close parentheses end fraction end style

= begin mathsize 16px style fraction numerator 2 open parentheses 1 plus sinA close parentheses over denominator cosA open parentheses 1 plus sinA close parentheses end fraction end style

= begin mathsize 16px style 2 over cosA equals 2 secA end style

Answer 2

$\ {\bold\purple{\dfrac{Cos \ A}{1 + Sin \ A} + \dfrac{1 + Sin \ A}{Cos \ A} = 2 \ sec \ A}}$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\begin{gathered}\implies\sf \dfrac{Cos \ A}{ 1 + Sin \ A} + \dfrac{1 + Sin \ A}{Cos \ A} \\\\\\:\implies\sf\dfrac{ Cos^2 A + \Bigg[1 + Sin \ A \Bigg]^2}{\Bigg[1 + Sin \ A \Bigg] Cos \ A} \\\\\\:\implies\sf \dfrac{ Cos^2 \ A + 1 \ Sin^2 \ A + 2 \ Sin \ A}{\Big[1 + Sin \ A \Big] + Cos \ A}\\\\\\:\implies\sf \dfrac{ 2 + 2 \ Sin \ A}{Cos \ A \Big[ 1 + Sin \ A \Big]}\\\\\\:\implies\sf \dfrac{ 2}{Cos \ A} \\\\\\:\implies{\bold\purple{ 2 \ Sec \ A}}\end{gathered}$

$\qquad\qquad{\bold\pink{Hence \ Proved!!}}$