KöMaL 2012 B4452

Let $t$>0 be a real number, and let $T_i$ denote the sum $t^{i}+\frac{1}{t^{i}}$ (for all positive integers $i$). Prove that $T_{1}T_{2}T_{4}\ldots T_{2^{k-1}}=T_{1}+T_{3}+T_{5}+\ldots +T_{2^{k}-3}+T_{2^{k}-1}$ for all positive integers $k$.