Question

If $p=\sqrt{\frac{1-\sin x}{1+\sin x}}$, $q=\frac{(1-\sin x)}{(\cos x)}$, $r=\frac{\cos x}{(1+\sin x)}$, then find the relation between $p,q$ and $r$.

Answer 1

Compute the required relation:

The given value of

$p=\sqrt{\frac{1-\sin x}{1+\sin x}}$

$q=\frac{(1-\sin x)}{(\cos x)}$

$r=\frac{\cos x}{(1+\sin x)}$

First consider $p$and rationalize it

$$\begin{gathered} p=\sqrt{\frac{1-\sin x}{1+\sin x}} \\ p=\sqrt{\frac{1-\sin x}{1+\sin x}\times \frac{1-\sin x}{1-\sin x}} \\ =\sqrt{\frac{{(1-\sin x)}^2}{1-{\sin }^{2}x}} \\ =\sqrt{\frac{{(1-\sin x)}^2}{{\cos }^{2}x}} \\ =\frac{1-\sin x}{\cos x} \end{gathered}$$

Now consider $r$

$$\begin{gathered} r=\frac{\cos x}{1+\sin x} \\ =\frac{\cos x}{1+\sin x}\times \frac{1-\sin x}{1-\sin x} \\ =\frac{\cos x\left(1-\sin x\right)}{1-{\sin }^{2}x} \\ =\frac{\cos x(1-\sin x)}{{\cos }^{2}x} \\ =\frac{1-\sin x}{\cos x} \end{gathered}$$

Hence $p=q=r=\frac{1-\sin x}{\cos x}$