(1+sint)/(cost)-(cost)/(1-sint)

Question

asked Feb 19, 2023 7.1k views

1 Answer

Slashdottir · Answer 1 · 2023-02-25T19:37:54+0000

$(1+\sin(t))/(\cos(t)) - (\cos(t))/(1-\sin(t))$

Multiply through the second term by
$1+\sin(t)$ :

$(1+\sin(t))/(\cos(t)) - (\cos(t))/(1-\sin(t))\cdot(1+\sin(t))/(1+\sin(t))$

$(1+\sin(t))/(\cos(t)) - (\cos(t)(1+\sin(t)))/(1-\sin^2(t))$

Recall that
$\cos^2(x)+\sin^2(x)=1$ for all x :

$(1+\sin(t))/(\cos(t)) - (\cos(t)(1+\sin(t)))/(\cos^2(t))$

If
$\cos(t)\\eq0$ , we can cancel a factor of it in the second fraction.

$(1+\sin(t))/(\cos(t)) - (1+\sin(t))/(\cos(t))$

and this simplifies to

$((1+\sin(t))-(1+\sin(t)))/(\cos(t)) = \boxed{0}$