Question

What is the value of \[\cot x - \tan x\]?
A. \[\cot 2x\]
B. \[2\cot 2x\]
C. \[2cot^{2}x\]
D. \[cot^{2}\left( {2x} \right)\]

Answer 1

Hint: In the given question, one trigonometric expression is given. First we will convert \[\cot x\] into \[\tan x\]. Then multiply 2 with the denominator and numerator. Then apply the formula \[\tan 2A = \dfrac{{2\tan A}}{{1 - {{\tan }^2}A}}\] to get the final result.

Formula used:
\[\cot x = \dfrac{1}{{\tan x}}\]
\[\tan 2A = \dfrac{{2\tan A}}{{1 - \tan^{2}A}}\]

Complete step by step solution:
The given trigonometric expression is \[\cot x - \tan x\].
Let’s simplify the given expression.
\[\cot x - \tan x = \dfrac{1}{{\tan x}} - \tan x\]
\[ \Rightarrow \]\[\cot x - \tan x = \dfrac{{1 - \tan^{2}x}}{{\tan x}}\]
Now multiply and divide the right-hand side of the above equation by 2.
\[\cot x - \tan x = \dfrac{{2\left( {1 - \tan^{2}x} \right)}}{{2\tan x}}\]
\[ \Rightarrow \]\[\cot x - \tan x = 2\left( {\dfrac{{1 - \tan^{2}x}}{{2\tan x}}} \right)\]
Now apply the formula \[\tan2A = \dfrac{{2\tan A}}{{1 - \tan^{2}A}}\].
\[\cot x - \tan x = \dfrac{2}{{\tan 2x}}\]
\[ \Rightarrow \]\[\cot x - \tan x = 2\cot 2x\]

Hence the correct option is option B.

Note: Students are often confused with the conversion of trigonometric functions. For this problem we used \[\cot x = \dfrac{1}{{\tan x}}\]. We know that, \[\tan x = \dfrac{{sinx}}{{cosx}}\] and \[\cot x = \dfrac{{cosx}}{{sinx}}\]. So, the reciprocal of \[\\tan x\] is \[\\cot x\].