Question

what is the formula of cot3A? ​

Answer 1

Answer:

prove that cot3A = 3cotA - cot^3A/1 - 3cot^2A.

Answer 2

The formula for $cot3A$ is $\frac{cot^{3}A - 3cotA }{3cot^{2}A - 1} .$

Given:

Given the trigonometrical term for which the formula is to be found,

That is, $cot3A$.

To Find:

We have to find the formula for $cot3A$.

Solution:

The trigonometrical formula for cot (A + B) is given by,

$Cot (A + B) = \frac{cotA.cotB - 1}{cot A + cot B}$.

Put A = B to get,

$Cot (2A) = \frac{cot^2A - 1}{2. cot A}$.

Therefore, formula for $cot3A$ is given by,

$cot 3A = cot (2A + A)$.

Using the formula of formula for cot (A + B), the above equation can be rewritten as,

$cot 3A = cot (2A + A) = \frac{cot2A.cotA - 1}{cot 2A + cot A}$

On substituting for $cot2A$, the above equation becomes,

$cot 3A = \frac{(\frac{cot^2A - 1}{2.cot A}) cotA - 1}{(\frac{cot^2A - 1}{2.cot A}) + cot A}$

$= \frac{\frac{cot^3A - cot A - 2.cotA}{2.cotA} }{\frac{cot^2A - 1 + 2.cot^2A}{2.cot A}}$

$= \frac{cot^{3}A - 3cotA }{3cot^{2}A - 1} .$

$i.e., cot 3A = \frac{cot^{3}A - 3cotA }{3cot^{2}A - 1} .$

Other expressions for $cot3A$ are:

$cot3A = \frac{cos3A}{sin3A}$ and

$cot3A = \frac{1}{tan3A}$.

Hence, the formula for $cot3A$ is $\frac{cot^{3}A - 3cotA }{3cot^{2}A - 1} .$

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