1(xiii)
Assertion (A): If $\sec\theta + \tan\theta = a$ and $\sec\theta - \tan\theta = b$ then $ab = 1$.
Reason (R): $\sec^2\theta - \tan^2\theta = 1$
- (a)(A) is true and (R) is false.
- (b)(A) is false and (R) is true.
- (c)Both (A) and (R) are true and (R) is the correct explanation of (A).
- (d)Both (A) and (R) are true, but (R) is not the correct explanation of (A).
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Given,
$$\begin{aligned}\Rightarrow \sec\theta + \tan\theta = a \\[1em] \Rightarrow \sec\theta - \tan\theta = b \\[1em] \Rightarrow ab = (\sec\theta + \tan\theta)(\sec\theta - \tan\theta) \\[1em] \Rightarrow ab = \sec^2\theta - \tan^2\theta \\[1em] \Rightarrow ab = 1\end{aligned}$$
So assertion (A) is true.
We know that,
⇒ $\sec^2\theta - \tan^2\theta = 1$
This is a fundamental trigonometric identity.
Thus, Both (A) and (R) are true and (R) is the correct explanation of (A).
Hence, option 3 is the correct option.