Question

If sec $\theta$ + tan $\theta$ = x, find the value of sec $\theta$.

Answer 1

$sec\theta +tan\theta =x----\left(1\right)\Rightarrow sec\theta =(x-tan\theta )$

Squaring both sides, we get,

$se{c}^2\theta =x^2-2xtan\theta +ta{n}^2\theta \Rightarrow se{c}^2\theta -ta{n}^2\theta =x^2-2xtan\theta \Rightarrow 1=x^2-2xtan\theta \Rightarrow 2xtan\theta =x^2-1\Rightarrow tan\theta =\frac{x^2-1}{2x}$

Put this value in equation (1), we get

$sec\theta +\frac{x^2-1}{2x}=x\Rightarrow sec\theta =\frac{x^2-1}{2x}-x\Rightarrow sec\theta =\frac{2x^2-x^2+1}{2x}\Rightarrow sec\theta =\frac{x^2+1}{2x}$