If \(A = 2\sin^2x – \cos2x\), Find the Interval in Which A Lies
Question
If
\[ A = 2\sin^2x – \cos2x, \]
then \(A\) lies in the interval
(a) \([-1,3]\)
(b) \([1,2]\)
(c) \([-2,4]\)
(d) none of these
Solution
Using the identity
\[ \cos2x = 1 – 2\sin^2x \]
Substitute into the given expression:
\[ A = 2\sin^2x – (1 – 2\sin^2x) \]
\[ A = 4\sin^2x – 1 \]
Since
\[ 0 \le \sin^2x \le 1, \]
multiplying by 4 gives
\[ 0 \le 4\sin^2x \le 4 \]
Subtracting 1 throughout,
\[ -1 \le A \le 3 \]
Therefore,
\[ A \in [-1,3] \]
Final Answer
\[ \boxed{[-1,3]} \]
Hence, the correct option is (a) \([-1,3]\).