a, b, c are 3 consecutive terms of an AP If tana, tanb, tanc( b ≠multiple. of .(π)/(2)) are also

step 1 I had is given that ABC has three conglary terms of an AP. Next, 10a, 10b, 10c, B is not a multi

A, b, c are 3 consecutive terms of an AP If tana, tanb,...

$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are 3 consecutive terms of an $\mathrm{AP}$ If $\tan \mathrm{a}, \tan \mathrm{b}, \tan c\left(\mathrm{~b} \neq \mathrm{multiple}\right.$ of $\left.\frac{\pi}{2}\right)$ are also in $\mathrm{AP}$, then (a) $\tan \mathrm{b}=2 \tan \mathrm{a}$ (b) $\tan a \times \tan c=\tan b$ (c) $\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}=0$ (d) $\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}$

$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are 3 consecutive terms of an $\mathrm{AP}$ If $\tan \mathrm{a}, \tan \mathrm{b}, \tan c\left(\mathrm{~b} \neq \mathrm{multiple}\right.$ of $\left.\frac{\pi}{2}\right)$ are also in $\mathrm{AP}$, then
(a) $\tan \mathrm{b}=2 \tan \mathrm{a}$
(b) $\tan a \times \tan c=\tan b$
(c) $\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}=0$
(d) $\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}$

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Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This means that for any three consecutive terms, a, b, and c, the relation b - a = c - b holds, making b the arithmetic mean of a and c.

Tangent Function

The tangent function is a trigonometric function defined as the ratio of the sine to the cosine of an angle. It has a period of ? and is not defined at angles where the cosine is zero, leading to vertical asymptotes. Its properties and behavior under addition and subtraction of angles are central to many trigonometric identities.

Relationship Between Functions and Their Arguments in AP

When the angles (or any quantities) are in arithmetic progression, it is interesting to analyze whether applying a function to these quantities preserves an arithmetic structure in the resulting values. In particular, exploring the conditions under which the tangent of angles in an arithmetic progression also forms an arithmetic progression imposes specific constraints on the possible values of the angles or the function values.

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