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प्रश्न
If A = cos2θ + sin4θ for all values of θ, then prove that `3/4` ≤ A ≤ 1.
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उत्तर
We have A = cos2θ + sin4θ
= cos2θ + sin2θ sin2θ ≤ cos2θ + sin2θ
Therefore, A ≤ 1
Also, A = cos2θ + sin4θ
= (1 – sin2θ) + sin4θ
= `(sin^2theta - 1/2)^2 + (1 - 1/4)`
= `(sin^2theta - 1/2)^2 + 3/4 > 3/4`
Hence, `3/4 < "A" < 1`.
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