Question

Solve: 2 + = 2. Give solutions in interval [0, 2)

Answer 1

`\because\cos 2x+\sin x=\sin 2x`
`\because\sin 2x=2\sin xcosx`
`\because\cos 2x=\cos ^2x-\sin ^2x`

→ Substitute cos 2x and sin 2x by their expressions

`\therefore\cos ^2x-\sin ^2+\sin x=2\sin x\cos x`

→ Subtract both sides by sin x

`\cos ^2x-\sin ^2x=2\sin xcosx-\sin x`

→ Take sin x as a common factor in the right side

`\therefore\cos ^2x-\sin ^2x=\sin x(2\cos x-1)`

From the graph, the equation has 4 solutions, the intersection points between the 2 graphs