Question

How do you solve \[\cos 2x=\sin x+\cos x\] on the interval \[\left[ 0,2pi \right]\]?

Answer 1

Hint: In this problem, we have to solve for x from the trigonometric expression within the given interval. To solve these types of problems, we have to know trigonometric formulas and rules. Here, we can find the x value by simplifying the given trigonometric expression using some trigonometric formulas.

Complete step by step answer:
We know that the given trigonometric expression is,
\[\cos 2x=\sin x+\cos x\]
We can bring the terms in right-hand side to the left-hand side, we get
\[\Rightarrow \cos 2x-\sin x-\cos x=0\]
We know that \[\cos 2x={{\cos }^{2}}x-{{\sin }^{2}}x\], applying this in the above step, we get
\[\Rightarrow {{\cos }^{2}}x-{{\sin }^{2}}x-\sin x-\cos x=0\]
Now we can write the first two terms in difference of square formula, we get
\[\Rightarrow \left( \cos x-\sin x \right)\left( \cos x+\sin x \right)-\left( \cos x+\sin x \right)=0\]
Now we can take the common terms outside, we get
\[\Rightarrow \left( \cos x+\sin x \right)\left( \cos x-\sin x-1 \right)=0\]……… (1)
\[\cos x+\sin x=0\]and \[\cos x-\sin x-1=0\]
From the above expression (1), we can take the first part to solve for x.
\[\begin{align}
  & \Rightarrow \cos x+\sin x=0 \\
 & \Rightarrow \cos x=-\sin x \\
 & \Rightarrow -1=\dfrac{\sin x}{\cos x} \\
 & \Rightarrow \tan x=-1\text{ }\because \text{tanx=}\dfrac{\sin x}{\cos x} \\
 & \Rightarrow x=135,315 \\
\end{align}\]
We know that tan value becomes -1 when x value is 135, 315
Therefore, x = 135, 315.
From the expression (2) we can take the second part to solve for x.
\[\begin{align}
  & \Rightarrow \cos x-\sin x-1=0 \\
 & \Rightarrow \cos x-\sin x=1 \\
 & \\
\end{align}\]
We can square on both sides to get,
\[\begin{align}
  & \Rightarrow {{\left( \cos x-\sin x \right)}^{2}}={{1}^{2}} \\
 & \Rightarrow {{\cos }^{2}}x+{{\sin }^{2}}x-2\sin x\cos x \\
 & \Rightarrow 1-2\sin x\cos x=1\text{ }\because {{\cos }^{2}}x+{{\sin }^{2}}x=1 \\
\end{align}\]
Now we can subtract -1 on both sides we get,
\[\Rightarrow -2\sin x\cos x=0\]
Now we can multiply -1 on both sides and simplify it, we get
\[\begin{align}
  & \Rightarrow 2\sin x\cos x=0 \\
 & \Rightarrow \sin 2x=0\text{ }\because \text{sin2x=2sinxcosx} \\
 & \Rightarrow 2x=0,180,360 \\
 & \Rightarrow x=0,90,180 \\
\end{align}\]
We know that sin value becomes 0, when x value is 0, 90, 180,
Therefore, x = 0, 90, 135, 180, 315.

Note:
 Students make mistakes while calculating the degree part, Students should always remember the degree values to solve these types of problems. Students should also concentrate on the trigonometric formula part to solve these types of problems. In these problems, we have used many formulas which should be remembered to solve these types of problems.