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Notation - $$\prod_{i=1}^{k} x_i = x_1 x_2 \cdots x_k$$. If $$x = \prod_{r=1^\circ}^{89^\circ} \log_e \tan r^\circ$$ and $$y = \prod_{r=1^\circ}^{89^\circ} \log_e \cot r^\circ$$ then $$\cos(x^2 - y^2 + 3xy)$$ is equal to
The product for $$x$$ contains the factor with $$r = 45^\circ$$, which is $$\log_e \tan 45^\circ = \log_e 1 = 0$$, so $$x = 0$$. Similarly $$\log_e \cot 45^\circ = 0$$, so $$y = 0$$. Hence $$\cos(x^2 - y^2 + 3xy) = \cos 0 = 1$$.
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