How do you simplify the expression $1 + \tan^{2} x$ ?

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Solution

Simplifying $1 + \tan^{2} x$ :
We know that $\tan x = \frac{\sin x}{\cos x}$ .
Thus, on substituting $\tan x = \frac{\sin x}{\cos x}$ in $1 + \tan^{2} x$ , we get:
$1 + \tan^{2} x = 1 + \frac{\sin^{2} x}{\cos^{2} x} = \frac{\cos^{2} x + \sin^{2} x}{\cos^{2} x} = \frac{1}{\cos^{2} x} [∵ \cos^{2} x + \sin^{2} x = 1] = s e c^{2} x [∵ \frac{1}{\cos x} = s e c x]$
Therefore, $1 + \tan^{2} x = s e c^{2} x$ .

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