The value of \(2\sin\theta\) can be \(a+\dfrac{1}{a}\), where \(a\gt0\) and \(a\ne1\).
False.
Step 1: We are told that \(a > 0\) and \(a \neq 1\).
Step 2: From the AM–GM inequality (Arithmetic Mean – Geometric Mean), we know that for any positive number \(a\):
\[ a + \dfrac{1}{a} \geq 2, \]
and equality (i.e., exactly equal to 2) happens only when \(a = 1\).
Step 3: Since the problem says \(a \neq 1\), this means:
\[ a + \dfrac{1}{a} > 2. \]
Step 4: Now, let’s check the possible values of \(2\sin\theta\).
We know that \(-1 \leq \sin\theta \leq 1\).
So multiplying the whole inequality by 2 gives:
\[ -2 \leq 2\sin\theta \leq 2. \]
Step 5: This means that \(2\sin\theta\) can never be greater than 2. Its maximum possible value is exactly 2, and minimum is -2.
Step 6: But from Step 3, \(a + \dfrac{1}{a} > 2\).
So \(a + \dfrac{1}{a}\) is always bigger than the maximum possible value of \(2\sin\theta\).
Final Step: Therefore, \(2\sin\theta\) can never equal \(a + \dfrac{1}{a}\) (except at \(a = 1\), which is not allowed).
Hence, the statement is False.