If sec θ + tan θ = x, write the value of sec θ − tan θ in terms of x.
Question:

If $\sec \theta+\tan \theta=x$, write the value of $\sec \theta-\tan \theta$ in terms of $x .$

Solution:

Given: $\sec \theta+\tan \theta=x$

We know that,

$\sec ^{2} \theta-\tan ^{2} \theta=1$

Therefore,

$\sec ^{2} \theta-\tan ^{2} \theta=1$

$\Rightarrow \quad(\sec \theta+\tan \theta)(\sec \theta-\tan \theta)=1$

$\Rightarrow \quad x(\sec \theta-\tan \theta)=1$

$\Rightarrow \quad(\sec \theta-\tan \theta)=\frac{1}{x}$

Hence, $\sec \theta-\tan \theta=\frac{1}{x}$

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