Question

If $\cos A + \sin A = \sqrt 2 \cos A$, then $\cos A - \sin A$ is:
A.\[\sqrt 2 \sin A\]
B.\[2\sin A\]
C.\[ - \sqrt 2 \sin A\]
D.\[ - 2\sin A\]

Answer 1

Hint:We begin by squaring both sides of the given equation. Then, use the identity ${\cos ^2}a + {\sin ^2}a = 1$ and substitute the required values. Next, rearrange the equation such that we get an expression equal to \[{\left( {\cos A - \sin A} \right)^2}\]. Next, take the square root on both the sides to get the required answer.

Complete step-by-step answer:
We are given that, $\cos A + \sin A = \sqrt 2 \cos A$
We will square both sides of the given equation
${\left( {\cos A + \sin A} \right)^2} = {\left( {\sqrt 2 \cos A} \right)^2}$
Then, simplify the expression using the formula ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
Hence, we have
\[{\cos ^2}A + {\sin ^2}A + 2\sin A\cos A = 2{\cos ^2}A\]
Next, we know that ${\cos ^2}a + {\sin ^2}a = 1$ which implies ${\cos ^2}A = 1 - {\sin ^2}A$ and ${\sin ^2}A = 1 - {\cos ^2}A$
We will substitute the values of ${\cos ^2}A$ and ${\sin ^2}A$
Therefore, we have
 $
  {\cos ^2}A + {\sin ^2}A + 2\sin A\cos A = 2{\cos ^2}A \\
  1 - {\sin ^2}A + 1 - {\cos ^2}A + 2\sin A\cos A = 2{\cos ^2}A \\
$
We wish to make the formula \[{\left( {\cos A - \sin A} \right)^2}\], therefore rearrange the terms such that we get an expression equal to \[{\left( {\cos A - \sin A} \right)^2}\]
$
  2 - 2{\cos ^2}A = {\cos ^2}A + {\sin ^2}A - 2\sin A\cos A \\
  2\left( {1 - {{\cos }^2}A} \right) = {\left( {\cos A - \sin A} \right)^2} \\
$
Also, ${\sin ^2}A = 1 - {\cos ^2}A$, substitute the value and take square root of the equation, then we will get,
$
  2{\sin ^2}A = {\left( {\cos A - \sin A} \right)^2} \\
  \cos A - \sin A = \sqrt 2 \sin A \\
$
Hence, option A is the correct answer.

Note For these types of questions, students must remember the formulas and identities of trigonometry. The approach used in this question is that we find the value of ${\left( {\cos A + \sin A} \right)^2}$ by squaring the given condition and then use various trigonometry identities to find the value of \[{\left( {\cos A - \sin A} \right)^2}\] followed by taking square root of both sides.