Question

Cos a/1-sin a + cos a/1+sin a =2 sec a

Answer 1

Answer:

To Prove:

Cos A/1-Sin A +Cos A/1+Sin A=Sec A

Proof:

=[Cos A(1-Sin A) +Cos A(1+Sin A)] / (1-Sin A)(1+Sin A)

=[(CosA-SinACosA)+(CosA+SinA CosA)] / 1-Sin^2A

=2CosA / Cos^2A

=2 / CosA

=2SecA

Answer 2

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

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

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

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