Solve trig equation

is it 2sin 2theta or 2 sin^2 theta?

Solve the following equations for 0 <= theta <= 360
i) 2sin2theta = costheta - I got values for theta as 90, 270, 30, 150.
however the answers in the book say: theta= 14.5, 90, 165.5, 270

$\begin{aligned} 2 \sin 2 \theta = \cos \theta & \Leftrightarrow 4\sin \theta \cos \theta - \cos \theta = 0 \\ & \Leftrightarrow \cos \theta (4\sin \theta - 1) = 0\end{aligned}$
So, $\cos \theta = 0$ or $4\sin \theta -1= 0$ and solve these.

Do the second question similarly.

there you go that's the way they got their answers if you don't understand a part you can ask😊

First of all, you're method is somewhat flawed (at least in first image since I can't be bothered to look at the second) because you make the rookie mistake of dividing through by $\cos \theta$ when you were supposed to factor it out instead. Then you found the correct solutions of 14.5 and 165.5 but to me it looks like you thought "Well, these are two out of four solutions mentioned, and I don't know where I'm supposed to get 90 and 270 from but I'll put them in there anyway!"

In fact, 90 and 270 come precisely from you factoring out $\cos \theta$ and setting it equal to 0

Secondly, please don't post full solutions on this forum - read the posting guidelines to provide effective help.