how do i rewrite as a algebraic expression

Question

please rewrite this trigonometric expression $\cos(\tan^{-1}(u)+\sin^{-1}(v))$ as an algebraic expression? I tried the sum differences method and could get no further

Answer 1

Hint

use these formulas

• $$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$

• $$1+\tan^2(X)=\frac{1}{\cos^2(X)}$$

Answer 2

Hint:

Set $\theta=\arctan u$, $\varphi=\arcsin v$. Use the addition formula for $\;\cos(\theta+\varphi)$.

This will require you to compute $\cos\theta$, $\sin\theta$ and $\cos\varphi$, $\sin \varphi$. We can easily deduce the squares of these numbers.

For instance, $\cos^2\theta=\dfrac1{1+\tan^2\theta}=\dfrac1{1+u^2},\;$ so $\;\cos \theta=\dfrac{\pm1}{\sqrt{1+u^2}}$. To determine the sign, you must remember the definition: $$\theta=\arctan u\stackrel{\text{def}}{\iff}\tan\theta=u\;\textbf{and}\;-\frac\pi2<\theta<\frac\pi2, $$ so the sign is +. Then you have also $\sin \theta$ at once: $$\sin\theta=\tan\theta\cos\theta=\frac u{\sqrt{1+u^2}}.$$ Do similarly for $\varphi$.