I have a problem with the question posted below:
I have been trying for a long time but I'm unable to simply this expression. The $(a+b)(a-b) $ like term gives us some hint but I'm not able to continue this pattern.
Would someone please help me to solve this question? Thanks for help.
Hint:
Use $\cos2t=2\cos^2t-1$
$$(2\cos t+1)(2\cos t-1)=\cdots=2(1+\cos2t)-1$$
$$f_n(t)=2\cos2^nt+1$$
Another way:
$2\cos2x+1=2(1-2\sin^2x)+1=\dfrac{\sin3x}{\sin x}$ for $\sin x\ne0$
Similarly $2\cos2x-1=\dfrac{\cos3x}{\cos x}$ for $\cos x\ne0$