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Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities1 Mark
Question
Write 'True' or 'False' and justify your answer in the following:
$\cos\theta=\frac{\text{a}^2+\text{b}^2}{2\text{ab}}$, where a and b are two lab distinct numbers such that ab > 0.
✓

Answer

False.
Given, a and b are two distinct numbers such that ab > 0.
Using, AM > GM
[since, AM and GM of two number a and b are $\frac{\text{a}+\text{b}}{2}$ and $\sqrt{\text{ab}},$ respectively]
$\Rightarrow\ \frac{\text{a}^2+\text{b}^2}{2}>\sqrt{\text{a}^2\times\text{b}^2}$
$\Rightarrow\ \text{a}^2+\text{b}^2>2\text{ab}$
$\Rightarrow\ \frac{\text{a}^2+\text{b}^2}{2\text{ab}}>1\ \Big[\because\ \cos\theta=\frac{\text{a}^2+\text{b}^2}{2\text{ab}}\Big]$
$\Rightarrow\ \cos\theta>1\ \big[\because\ -1\leq\cos\theta\leq1\big]$
Which is not possible.
Hence, $\cos\theta\neq\frac{\text{a}^2+\text{b}^2}{2\text{ab}}$

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