Question

I need help proving trigonometric identities[(1+sinθ)÷cosθ]+[cosθ÷(1+sinθ)] is identical to 2secθ

Answer 1

$(\frac{1+sin}{1 + sin})\frac{1 + sin}{cos} + (\frac{cos}{cos} )\frac{cos}{1 + sin} = 2 sec$

$\frac{1 + 2sin + sin^{2} }{cos(1 + sin)} + \frac{cos^{2} }{cos(1 + sin)} = 2 sec$

$\frac{1 + 2sin + sin^{2} + cos^{2}}{cos(1 + sin)} = 2 sec$

$\frac{1 + 2sin + 1}{cos(1 + sin)} = 2 sec$

$\frac{2 + 2sin}{cos(1 + sin)} = 2 sec$

$\frac{2(1 + sin)}{cos(1 + sin)} = 2 sec$

$\frac{2}{cos} = 2 sec$

   2 secθ = 2 secθ

Answer 2

'Ill use s for sin and c for cos

(1+s) / c  + c / (1+x)

= [(1+s)(1+s) + c^2] / c(1+s)

=  1 + 2s + s^2 + c^2  / c(1+s)

= 2+2s / c(1+s)       (because s^2 + c^2 = 1}

= 2(1+s) / c(1+s)

= 2/c

= 2 / cos θ

= 2 sec θ   answer