Question

Write the trigonometric expression as an algebraic expression sin(arctanx+arcsinx)

Answer 1

Treat \(\arctan x\) and \(\arcsin x\) as angle:
\[\sin(\arctan x+\arcsin x)=\sin(\arctan x)\cos(\arcsin x)+\cos(\arctan x)\sin(\arcsin x)\]
(This is the angle sum identity for sine.)
Now let's write the individual angles like this:
\[\begin{cases}
\theta=\arctan x\\\phi=\arcsin x
\end{cases}~~\implies~~\begin{cases}\tan\theta=x\\\sin\phi=x\end{cases}\]
You can use the trig ratios on the right to set up some right triangles, and solve for th missing side.

Tangent gives opposite over adjacent, so you can write this as \(\tan\theta=\dfrac{x}{1}\), and sine give opposite over hypotenuse, so \(\sin\phi=\dfrac{x}{1}\).

Your triangles:

Note that \(\phi\) isn't necessarily larger than \(\theta\); I just wanted to get across that the angles are different (though it's possible they may be equal).