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anemone
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For real numbers \(\displaystyle 0\lt x\lt \frac{\pi}{2}\), prove that $\cos^2 x \cot x+\sin^2 x \tan x\ge 1$.
anemone said:For real numbers \(\displaystyle 0\lt x\lt \frac{\pi}{2}\), prove that $\cos^2 x \cot x+\sin^2 x \tan x\ge 1$.
lfdahl said:My suggested solution:
By the AM-GM inequality, we get:
$\frac{\cos^3x}{\sin x}+\frac{sin^3x}{\cos x} \ge 2\sqrt{\frac{\cos^3x\sin^3x}{\cos x\sin x}} = \sin 2x \le 1, \;\;\; 0 < x < \frac{\pi}{2}$
Thus
$\cos^2x\cot x+\sin^2x\tan x \ge 1$
kaliprasad said:I aam sorry if I misunderstood but
$a > b $ and $ b <=c$ does not mean $ a > c$
A trigonometric inequality is an inequality that involves trigonometric functions, such as sine, cosine, and tangent. It is used to compare the values of different trigonometric expressions.
The most common trigonometric inequalities are sine, cosine, and tangent inequalities. These include sin x ≤ 1, cos x ≤ 1, and tan x < 1, among others.
To solve a trigonometric inequality, the same methods used to solve regular algebraic inequalities are used. This includes using algebraic manipulation, factoring, and graphing to find the solution set.
Trigonometric inequalities are used in various fields, such as physics, engineering, and economics, to model and solve real-world problems involving angles and distances.
While a calculator can be used to check the solutions of trigonometric inequalities, it cannot be used to solve them. Solving trigonometric inequalities requires understanding of trigonometric identities and properties.