Question

If \[\tan x = \dfrac{b}{a}\], then what is the value of the trigonometric expression \[a \cos 2x + b \sin 2x\]?
A. \[a\]
B. \[a - b\]
C. \[a + b\]
D. \[b\]

Answer 1

Hint: Simplify the trigonometric equation using the formulas of \[\cos 2x\] and \[\sin 2x\] in terms of \[\tan x\]. After that, substitute the value of \[\tan x\] in the equation and simplify it to get the required answer.

Formula Used:
\[\sin 2x = \dfrac{{2\tan x}}{{1 + \tan^{2}x}}\]
\[\cos 2x = \dfrac{{1 - \tan^{2}x}}{{1 + \tan^{2}x}}\]

Complete step by step solution:
The given trigonometric expression is \[a \cos 2x + b \sin 2x\], and \[\tan x = \dfrac{b}{a}\].
Let’s simplify the given expression.
\[f\left( x \right) = a \cos 2x + b \sin 2x\]
Substitute the formulas of \[\cos 2x\] and \[\sin 2x\] in terms of \[\tan x\].
\[f\left( x \right) = a \left( {\dfrac{{1 - \tan^{2}x}}{{1 + \tan^{2}x}}} \right) + b \left( {\dfrac{{2\tan x}}{{1 + \tan^{2}x}}} \right)\]
\[ \Rightarrow \]\[f\left( x \right) = \dfrac{{a\left( {1 - \tan^{2}x} \right) + 2b\tan x}}{{1 + \tan^{2}x}}\]
Substitute \[\tan x = \dfrac{b}{a}\] in the above equation.
\[f\left( x \right) = \dfrac{{a\left( {1 - {{\left( {\dfrac{b}{a}} \right)}^2}} \right) + 2b\left( {\dfrac{b}{a}} \right)}}{{1 + {{\left( {\dfrac{b}{a}} \right)}^2}}}\]
Simplify the equation.
\[f\left( x \right) = \dfrac{{a\left( {\dfrac{{{a^2} - {b^2}}}{{{a^2}}}} \right) + 2\left( {\dfrac{{{b^2}}}{a}} \right)}}{{\dfrac{{{a^2} + {b^2}}}{{{a^2}}}}}\]
\[ \Rightarrow \]\[f\left( x \right) = \dfrac{{\dfrac{{{a^2} - {b^2}}}{a} + \dfrac{{2{b^2}}}{a}}}{{\dfrac{{{a^2} + {b^2}}}{{{a^2}}}}}\]
Add the terms of the numerator with the common denominator.
\[f\left( x \right) = \dfrac{{\dfrac{{{a^2} - {b^2} + 2{b^2}}}{a}}}{{\dfrac{{{a^2} + {b^2}}}{{{a^2}}}}}\]
\[ \Rightarrow \]\[f\left( x \right) = \dfrac{{{a^2} + {b^2}}}{{\left( {\dfrac{{{a^2} + {b^2}}}{a}} \right)}}\]
Cancel out the common terms.
\[f\left( x \right) = \dfrac{1}{{\left( {\dfrac{1}{a}} \right)}}\]
\[ \Rightarrow \]\[f\left( x \right) = a\]

Hence the correct option is A.

Note: Euler's formula and the Binomial theorem are used to calculate the multiple angles of the type \[\sin nx\], \[\cos nx\], and \[\tan nx\].
The multiple angle formulas for the basic trigonometric functions are:
\[\sin nx = \sum\limits_{k = 0}^n {\cos^{k}} x \sin^{\left( {n - k} \right)}x \sin\left[ {\dfrac{\pi }{2}\left( {n - k} \right)} \right]\]
\[\cos nx = \sum\limits_{k = 0}^n {cos^{k}} x \sin^{\left( {n - k} \right)}x \cos\left[ {\dfrac{\pi }{2}\left( {n - k} \right)} \right]\]
\[\tan nx = \dfrac{{\sin nx}}{{\cos nx}}\]