Show that when written in terms of $ t$, where $t = \tan(x/2)$, the expression $2(1 + \cos(x))(5\sin(x) + 12\cos(x) + 13)$ is a perfect square.

Question

Attempt: I've used that $\sin(x) = (2t)/(1+t^2)$ and $\cos(x) = (1-t^2)/(1+t^2)$. However I don't seem to get a perfect square, instead I get $$(2/((1+t^2)^2))(14t^2 +20t + 38)$$.

I'm not sure if there error is with my method or my workings.

Any help would be greatly appreciated.

Answer 1

$$\begin{align} 2(1+\cos(x))(5\sin(x)+12\cos(x)+13)&=2\left(1+\frac{1-t^2}{1+t^2}\right)\left(\frac{10t}{1+t^2}+\frac{12-12t^2}{1+t^2}+13\right)\\ &=\frac{4}{(1+t^2)^2}(t+5)^2 \end{align}$$

Answer 2

I get $$\frac{4 {{\left( t+5\right) }^{2}}}{{{\left( {{t}^{2}}+1\right) }^{2}}}$$

Answer 3

If $x= \pi/4$, then compute that

$2(1+ \cos x)(5 \sin x+ 12 \cos x+13)$ is not a natural number !