Question

A student solve the equation… (question in the picture). Thank you

Answer 1

The equation is given below as

`(sin(2x))/(cos(x))=2,0\leq x\leq\pi`

Step 1:

Cross multiply

`\begin{gathered} \begin{equation*} (sin(2x))/(cos(x))=2 \end{equation*} \\ sin(2x)=2cos(x) \\ sin(2x)-2cos(x)=0 \end{gathered}`

Step 2:

Apply the trig identity below

`sin(2x)=2sin(x)cos(x)`
`\begin{gathered} s\imaginaryI n(2x)-2cos(x)=0 \\ 2sin(x)cos(x)-2cos(x)=0 \\ by\text{ rearranging, we will have} \\ -2cos(x)+2sin(x)cos(x)=0 \end{gathered}`

Step 3:

Factor out 2cos(x)

`\begin{gathered} -2cos(x)+2s\imaginaryI n(x)cos(x)=0 \\ 2cos(x)(-1+sin(x))=0 \\ 2cos(x)(sin(x)-1)=0 \\ 2cos(x)=0,sin(x)-1=0 \\ cos(x)=0,sin(x)=1 \\ x=\cos^(-1)0,x=\sin^(-1)1 \\ x=(\pi)/(2),x=(\pi)/(2), \end{gathered}`

Step 4:

Check for undefined points

`\begin{gathered} cos(x)=0 \\ x=\cos^(-1)0 \\ x=(\pi)/(2) \end{gathered}`

Since the equation is undefined for π/2,

Hence,

There is no solution for x

`\mathrm{No\:Solution\:for}\:x\in \mathbb{R}`

Her error is that π/2 will make the equation undefined for the equation in the question.

Therefore,

π/2 cannot be the answer