Simplify:
\( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \)
Multiply numerator and denominator by \( 1 + \sin x \):
\[ \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \cdot \frac{1 + \sin x}{1 + \sin x} \right) = \tan^{-1} \left( \frac{\cos x (1 + \sin x)}{(1 - \sin x)(1 + \sin x)} \right) \]
Denominator becomes:
\[ (1 - \sin x)(1 + \sin x) = 1 - \sin^2 x = \cos^2 x \]
Numerator becomes:
\[ \cos x (1 + \sin x) \]
So the expression becomes:
\[ \tan^{-1} \left( \frac{\cos x (1 + \sin x)}{\cos^2 x} \right) \]
Simplify:
\[ \tan^{-1} \left( \frac{1 + \sin x}{\cos x} \right) \]
Now, use identity:
\[ \frac{1 + \sin x}{\cos x} = \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) \]
Therefore,
\[ \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) = \frac{\pi}{4} + \frac{x}{2} \]
The standard electrode potential for Ni²⁺/Ni couple is -0.25 V and for Ag⁺/Ag couple is 0.80 V. These two couples are connected to make an electrochemical cell. The redox reaction is spontaneous. The cell potential will be: