Evaluate
$$\cos 10^{\circ}+\cot 23^{\circ}-\sin 12^{\circ} + cos 12^{\circ}$$
Can you help me please!
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@coffeemath it does actually equal 36 – 5201314 Feb 15 '21 at 16:45
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@5201314 My bad. was using secant. Erasing comment. – coffeemath Feb 15 '21 at 16:48
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See https://math.stackexchange.com/questions/175736/evaluate-tan220-circ-tan240-circ-tan280-circ – lab bhattacharjee Feb 15 '21 at 16:59
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2@Liza Please don't retroactively erase your question after it already has an accepted answer. I've rolled it back to what seems to be the most recent valid question. – Erick Wong Feb 17 '21 at 21:55
1 Answers
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Per the identity $4\cos^3t-3\cos t=\cos 3t$, verify that $\cos20$, $-\cos40$ and $-\cos80$ are the three roots of $$4x^3-3x=\frac12 $$ Square to get $64x^6-96x^4+36x^2-1=0$, or the cubic equation for $\frac1{x^2}$
$$\frac1{x^6}-\frac{36}{x^4} +\frac{96}{x^2} -64=0$$
Thus $$ \csc^2 10+\csc^2 50+\csc^2 70= \frac1{\cos^280}+ \frac1{\cos^240}+ \frac1{\cos^220}=36 $$
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