\(f(x) = \frac{1}{\tan x+\cot x},\) ప్రమేయాల యొక్క గరిష్ట విలువ ఎంత, ఇక్కడ \(0 < x < \frac{\pi}{2}?\)

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NDA 01/2022: Maths Previous Year paper (Held On 10 April 2022)

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Answer 1

ఉపయోగించిన సూత్రం:

గణన:

\(f(x) = \frac{1}{\tan x+\cot x},\) \(0 < x < \frac{\pi}{2}?\)

⇒ \(f(x) = \frac{1}{\tan x+\cot x},\)

⇒ \(f(x) = \frac{1}{\frac{sin x}{cosx}+\frac{cosx}{sinx}}\)

⇒ \(f(x) =\frac{sinx\ cosx}{sin^2x+cos^2x}\)

⇒ f(x) = sin x cos x (\(\frac{2}{2}\)) [∵ sin2θ + cos2θ = 1]

⇒ f(x) = \(\frac{1}{2}\)sin 2x [∵ 2sin θ cosθ = sin 2θ]

మనకు తెలుసు, -1 ≤ sin θ ≤ 1

⇒ -1 ≤ sin 2x ≤ 1

∴ f(x)max = \(\frac{1}{2}\)(1) = 1/2