# Showing that $\sin x\;f(\sin x)\;f^\prime(\cos x)+\cos x\;f(\cos x)\;f^\prime(\sin x)=\frac{2}{\pi\sin x\cos x}$ for $f(x)$ defined by a series

Let $$f(x) = 1 + \left(\frac12\cdot x\right)^2+\left(\frac12\cdot\frac34 \cdot x^2\right)^2+\left(\frac12\cdot\frac34\cdot\frac56\cdot x^3\right)^2+\cdots$$ Prove that $$\sin x\;f(\sin x)\;f^\prime(\cos x)+\cos x\;f(\cos x)\;f^\prime(\sin x)=\frac{2}{\pi\sin x\cos x}$$

No hints are available. Any help would be appreciated. Thanks.

I have been trying to solve this problem by finding explicit form of $$f(x)$$. I have noticed that the series is similar to the power series of $$1/\sqrt{1-x}$$, thus I have spent a lot of time connecting it to $$f(x)$$. Apparently, it did not work.

I also have tried to transform the claimed equality into well-known identity of trigonometric functions, such as $$\sin^2x+\cos^2x =1$$.

(Edited) Thanks Parcly Taxel for editing the article. I did not know how to ask properly.

• What's the source of this problem? Is the expectation that the problem requires some sophisticated theorems, or that it becomes trivial by invoking "one cool trick"? – Blue Oct 28 '18 at 11:43
• It is fairly easy to see that $LHS=f(\sin x)f(\cos x) \frac{d}{dx} \log \frac{f(\sin x)}{f(\cos x)}$. Just not sure what's next. – AdditIdent Oct 28 '18 at 12:09
• As requested by @Blue you should mention the source of the problem. I also saw a comment giving link to the source of the problem which was later deleted. Anyway the link is now a part of my answer and should help add context to your problem. – Paramanand Singh Oct 28 '18 at 14:21

We have $$f(x) = \frac{2K(x)}{\pi}\tag{1}$$ where $$K(x)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-x^2\sin^2t}},E(x)=\int_{0}^{\pi/2}\sqrt{1-x^2\sin^2t}\,dt\tag {2}$$ Now Legendre's identity states that $$K(x) E(\sqrt {1-x^2})+K(\sqrt{1-x^2})E(x)-K(x)K(\sqrt{1-x^2})=\frac{\pi}{2}\tag{3}$$ We also need the formula $$E(x) =x(1-x^2) \frac{dK(x)} {dx} +(1-x^2) K(x)\tag{4}$$ Using $$(1)$$ and $$(4)$$ the identity $$(3)$$ can be transformed into an identity connecting $$f$$ and its derivative $$f'$$ and the finally replacing $$x$$ by $$\sin x$$ completes the job.