TS EAMCET · Maths · Trigonometric Ratios & Identities
If \(A=\sin ^2 \theta+\cos ^4 \theta\), then for all values of \(\theta, A\) lies in the interval
- A \([1,2]\)
- B \(\left[\frac{3}{4}, 1\right]\)
- C \(\left[\frac{1}{2}, \frac{3}{4}\right]\)
- D \(\left[\frac{3}{4}, \frac{19}{16}\right]\)
Answer & Solution
Correct Answer
(B) \(\left[\frac{3}{4}, 1\right]\)
Step-by-step Solution
Detailed explanation
We have, \[ \begin{aligned} A & =\sin ^2 \theta+\cos ^4 \theta \\ & =1-\cos ^2 \theta+\cos ^4 \theta \\ & =1-\cos ^2 \theta\left(1-\cos ^2 \theta\right) \\ & =1-\cos ^2 \theta \sin ^2 \theta \\ & =1-\frac{1}{4} \sin ^2 2 \theta \end{aligned} \] We know that,…
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