Assertion (A): $\sec θ(1 + \sin θ)(\sec θ + \tan θ) = 1$
Reason (R): $\sec^2 θ - \tan^2 θ = 1$, for any value of $θ$.
- (a)(A) is true, (R) is false.
- (b)(A) is false, (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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Solving the L.H.S. of the Assertion,
$$\begin{aligned}\Rightarrow \sec θ(1 + \sin θ)(\sec θ + \tan θ) \\[1em] \Rightarrow \sec θ(1 + \sin θ)\Big(\frac{1}{\cos θ} + \frac{\sin θ}{\cos θ}\Big) \\[1em] \Rightarrow \sec θ(1 + \sin θ) \times \frac{1 + \sin θ}{\cos θ} \\[1em] \Rightarrow \frac{1}{\cos θ} \times \frac{(1 + \sin θ)^2}{\cos θ} \\[1em] \Rightarrow \frac{(1 + \sin θ)^2}{\cos^2 θ} \\[1em] \Rightarrow \frac{(1 + \sin θ)^2}{1 - \sin^2 θ} \\[1em] \Rightarrow \frac{(1 + \sin θ)^2}{(1 - \sin θ)(1 + \sin θ)} \\[1em] \Rightarrow \frac{1 + \sin θ}{1 - \sin θ}.\end{aligned}$$
Since, $\frac{1 + \sin θ}{1 - \sin θ} \ne 1$, the Assertion (A) is false.
Now, $\sec^2 θ - \tan^2 θ = 1$ is a fundamental trigonometric identity which is true.
So, Reason (R) is true.
Thus, (A) is false, (R) is true.
Hence, option 2 is the correct option.