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I was doing basic excercises about proofs but I am lost with this one.

"Prove taht if a is a fixed value in the interval {$0<a<\pi$}, then for every number t in the interval {$ 0 \le t \le \pi$ -a}, then:

$$ F(x) = \frac{sin(t) + sin(t+a)}{cos(t) + cos(t+a)} $$ is an independent from t.

I will also leave the original text in spanish just in case I misstranslated something mostrar "marcha atrás" que si a es un número fijo que verifica {$0<a<\pi$}, entonces para cualquier número t que verifica {$ 0 \le t \le \pi$ -a}, resulta que $$ F(x) = \frac{sin(t) + sin(t+a)}{cos(t) + cos(t+a)} $$ es un valor independiente de t.

Javier Lázaro
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  • Do you mean $F(t)$? The right hand is not a function of $x$. And it is not a constant function of $t$. – lulu Mar 30 '24 at 14:39

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Using the formula of the sum of the sinus and cosinus:

$$ \sin(a) + \sin(b) = 2\sin(\frac{a+b}{2})\cos(\frac{a-b}{2}) $$ $$ \cos(a) + \cos(b) = 2\cos(\frac{a+b}{2})\cos(\frac{a-b}{2}) $$

With this in mind, we can simplify $F(t)$: $$ F(t) = \frac{2\sin(\frac{t+t+a}{2})\cos(\frac{t-t-a}{2})}{2\cos(\frac{t+t+a}{2})\cos(\frac{t-t-a}{2})} = \frac{\sin(t+\frac{a}{2})}{\cos(t+\frac{a}{2})} = \tan(t+\frac{a}{2}) $$

Which is not a constant value. Maybe you made some error while writing the problem?

A. Person
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