Question

If $0 <\alpha, \beta, \gamma < \dfrac{\pi }{2}$ such that \[\alpha + \beta + \gamma = \dfrac{\pi }{2}\] and \[\cot \alpha ,\cot \beta ,\cot \gamma \] are in A.P., then the value of \[\cot \alpha \cot \gamma \] is:

Answer 1

Hint: Arithmetic progression (AP) is a sequence whose terms increase or decrease by a fixed number. The fixed number is called the common difference. If ‘a’ is the first term and ‘d’ is a common difference, then AP can be written as a, a+d, a+2d,….a+(n-1)d,…..
For example, if x, y, z are in AP then it implies that x + z = 2y
Given: \[\alpha + \beta + \gamma = \dfrac{\pi }{2}\] and \[\cot \alpha ,\cot \beta ,\cot \gamma \] are in AP
To find: the value of \[\cot \alpha \cot \gamma \]

Complete step-by-step solution:
Step 1: As we know that if three numbers, let say x, y and z, are in AP then it implies that \[x + z = 2y\]
Now according to the question \[\cot \alpha ,\cot \beta ,\cot \gamma \] are in AP, therefore we can write
\[\cot \alpha + \cot \gamma = 2\cot \beta \]
Also, it is given that \[\alpha + \beta + \gamma = \dfrac{\pi }{2}\] , we can also write it as
\[\beta = \dfrac{\pi }{2} - (\alpha + \beta )\]
Now substituting the value of \[\beta \] in equation \[\cot \alpha + \cot \gamma = 2\cot \beta \] , we get
\[\cot \alpha + \cot \gamma = 2\cot \beta \]
\[\cot \alpha + \cot \gamma = 2\cot (\dfrac{\pi }{2} - (\alpha + \gamma ))\]
Step 2: Now from trigonometric properties we know that,
\[\cot (\dfrac{\pi }{2} - x) = \tan x\]
Therefore, we have
\[\cot \alpha + \cot \gamma = 2\cot (\dfrac{\pi }{2} - (\alpha + \gamma ))\]
\[\cot \alpha + \cot \gamma = 2\tan (\alpha + \gamma )\]
\[\dfrac{1}{{\tan \alpha }} + \dfrac{1}{{\tan \gamma }} = 2\tan (\alpha + \gamma )\]
Step 3: As we know that,
\[\tan (A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}\]
Using the above trigonometric formula, we have
\[\dfrac{1}{{\tan \alpha }} + \dfrac{1}{{\tan \gamma }} = 2\tan (\alpha + \gamma )\]
\[\dfrac{1}{{\tan \alpha }} + \dfrac{1}{{\tan \gamma }} = 2\dfrac{{\tan \alpha + \tan \gamma }}{{1 - \tan \alpha \tan \gamma }}\]
On further simplification we get
\[\dfrac{{\tan \alpha + \tan \gamma }}{{\tan \alpha \tan \gamma }} = 2\dfrac{{\tan \alpha + \tan \gamma }}{{1 - \tan \alpha \tan \gamma }}\]
Canceling the term \[\tan \alpha + \tan \gamma \] from both sides, we get
\[\dfrac{1}{{\tan \alpha \tan \gamma }} = 2\dfrac{1}{{1 - \tan \alpha \tan \gamma }}\]
\[1 - \tan \alpha \tan \gamma = 2\tan \alpha \tan \gamma \]
Taking the term \[\tan \alpha \tan \gamma \]on the right side, we get
\[1 = 2\tan \alpha \tan \gamma + \tan \alpha \tan \gamma \]
\[1 = 3\tan \alpha \tan \gamma \]
Now from trigonometric properties, we know that \[\cot x = \dfrac{1}{{\tan x}}\] , using this trigonometric property we can write
\[1 = 3\tan \alpha \tan \gamma \]
\[1 = 3\dfrac{1}{{\cot \alpha \cot \gamma }}\]
\[\cot \alpha \cot \gamma = 3\] (which is the required answer)
Hence, the value \[\cot \alpha \cot \gamma \] is equal to 3.

Note: Always use trigonometric properties to convert the given equation into one identity (tan or cot or sin or cos etc) and also it reduces the equation in the simplest form.
If three numbers a, b and c are in AP then it implies that \[a + c = 2b\].