What is the Conversion of Trigonometric Functions in Integration Problems?

[abot]

in the attachment there is a problem related to techniques of integration, i fully understand everything until near the end of the problem they answer says

(1/6)o - (1/12)sin2o + C = (1/6)o - (1/6)sino coso + C

and then they convert the sinocoso to sec-1

this part is really confusing...can you help me please...

thank you

 

[benorin]

Recall that $$\sin(2o)=2\sin (o)\cos (o)$$, this is the first one.

Note that, from the triangle, we have $$\sec (\theta ) = \frac{x}{3} \Rightarrow \theta = \sec ^{-1} \left( \frac{x}{3}\right)$$
and, also from the triangle, we have $$\cos (\theta ) = \frac{3}{x}$$

and $$\sin (\theta ) = \mbox{ what? }$$, and the required result follows.