Question

What do you get when ${{\left[ \tan \left( x \right) \right]}^{2}}$? Is it ${{\tan }^{2}}\left( x \right)$ or $\tan {{\left( x \right)}^{2}}$? Also is $\tan \left( \tan x \right)$ the same as ${{\tan }^{2}}\left( x \right)$?

Answer 1

For solving this question we will use the basic trigonometry functions and its applications. We will discuss some of the trigonometric function conditions and its properties and then we will correlate those discussed properties to our question and later we come to a conclusion and solve the given question. So, we proceed with our solution as follows.

Complete step by step solution:
Generally in trigonometry we have the functions or expressions like ${{\sin }^{2}}x,{{\cos }^{2}}x,{{\tan }^{2}}x$.
Theses expressions like ${{\sin }^{2}}x,{{\cos }^{2}}x,{{\tan }^{2}}x$ are used as shorthand notation for ${{\left( \sin x \right)}^{2}},{{\left( \cos x \right)}^{2}},{{\left( \tan x \right)}^{2}}$ respectively while solving the questions as it becomes easier for us to write and also for the understanding purpose of the solution.
Note that if conventions are not clear, then when we write $\tan {{x}^{2}}$ then we could intend that $\tan \left( {{x}^{2}} \right)$ or ${{\left( \tan \left( x \right) \right)}^{2}}$.
So generally in many questions the popular practice is to write ${{\tan }^{2}}x$ when we mean ${{\left( \tan \left( x \right) \right)}^{2}}$ and $\tan \left( {{x}^{2}} \right)$ when we mean $\tan \left( {{x}^{2}} \right)$.
$\Rightarrow \tan \left( \tan x \right)$ is not the same as ${{\tan }^{2}}\left( x \right)$ because ${{\tan }^{2}}\left( x \right)$ will be equal to $\left( \tan x \right)\left( \tan x \right)$.
Therefore we could conclude that ${{\left[ \tan \left( x \right) \right]}^{2}}$ is equal to ${{\tan }^{2}}\left( x \right)$ and ${{\left[ \tan \left( x \right) \right]}^{2}}$ is not equal to $\tan {{\left( x \right)}^{2}}$. The function $\tan \left( \tan x \right)$ is not the same as ${{\tan }^{2}}\left( x \right)$.

Note: Students must not be confused in between the functions like $\tan {{\left( x \right)}^{2}}$ and ${{\tan }^{2}}\left( x \right)$. We must be able to differentiate and find the difference in between these terms. We must have good knowledge in the concept of trigonometric functions and the functions properties to understand and solve these kind of questions.