~~Proving Trigonometric Functions~~ ♥

[yasaminG]

i have to prove that

Sin (s+t) / cos s cos t = tans + tan t

 

[soroban]

Hello, yasaminG!

You're expected to know:

$\displaystyle \sin(A\,+\,B)\:=\:\sin(A)\cdot\cos(B)\,+\,\sin(B)\cdot\cos(A)$ .and . $\displaystyle \frac{\sin A}{\cos A}\:=\:\tan A$

Prove: .$\displaystyle \frac{\sin(s\,+\,t)}{\cos(s)\cdot\cos(t)}\:=\:\tan(s)\,+\,\tan(t)$

The left side is: .\(\displaystyle \L\frac{\sin(s)\cdot\cos(t)\,+\,\sin(t)\cdot\cos(s)}{\cos(s)\cdot\cos(t)}\;=\;\frac{\sin(s)\cdot\cos(t)}{\cos(s)\cdot\cos(t)}\,+\,\frac{\sin(t)\cdot\cos(s)}{\cos(s)\cdot\cos(t)}\)

. . . \(\displaystyle \L=\:\frac{\sin(s)}{\cos(s)}\,+\,\frac{\sin(t)}{\cos(t)}\:=\:\tan(s)\,+\,\tan(t)\)