Question

1+SinA÷CosA + CosA÷1+SinA=2SecA

Answer 1

Answer:

LHS=1+sinA/cosA+cosA/1+sinA

=(1+sinA)^2+cos^2A/cosA(1+sinA)

=1+2sinA+sin^2A+cos^2A/cosA(1+sinA)

=1+2sinA+1/cosA(1+sinA)

=2+2sinA/cosA(1+sinA)

=2(1+sinA)/cosA(1+sinA)

=2/cosA

=2secA=RHS

Answer 2

Solution :-

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

Consider LHS

$\sf = \dfrac{1 +sinA}{cosA} + \dfrac{cosA}{1 + sinA}$

Taking LCM

$\sf = \dfrac{(1 +sinA)^{2} + cos^{2} A}{cosA(1 + sinA)}$

$\sf = \dfrac{1 + sin^{2}A + 2sin A + cos^{2} A}{cosA(1 + sinA)}$

[ Because (x + y)² = x² + y² + 2xy ]

Rearranging the terms in numerator

$\sf = \dfrac{1 +( sin^{2}A + cos^{2} A )+ 2sinA}{cosA(1 + sinA)}$

$\sf = \dfrac{1 +1+ 2sinA}{cosA(1 + sinA)}$

[ Because sin²A + cos²A = 1 ]

$\sf = \dfrac{2+ 2sinA}{cosA(1 + sinA)}$

$\sf = \dfrac{2(1 + sinA)}{cosA(1 + sinA)}$

$\sf = \dfrac{2}{cosA}$

It can be written as

$\sf = 2 \times \dfrac{1}{cosA}$

$\sf = 2 \times secA$

= RHS

LHS = RHS

Hence proved.