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I was doing a physics problem, on solving it with two different methods, I got $$F(Force) = mg\cos(Q)$$ and $$F=\frac{mg}{\sin(Q)}$$ ($Q$ is an angle), $m=$ mass of the body (constant), $g=$ gravitational acceleration (constant). $Q=\theta$(angle) Since the Forces are equal, this means $$\cos Q=\csc Q$$

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We note that $$ \cos Q = \csc Q $$ holds if and only if $$ \cos Q \sin Q = 1, $$ which holds if and only if $$ 2 \cos Q \sin Q = 2, $$ or $$ \sin 2Q = 2, $$ which is clearly impossible.

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    $\begingroup$ +1. Unless $Q$ is complex valued $\endgroup$ – Qurultay Feb 14 at 18:53
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Cosine never equals cosecant. First, the former must have absolute value $\le 1$, while the latter must have absolute value $\ge 1$. This means that if the cosine of an angle were equal to the cosecant, then they must either both equal $1$ or both equal $-1$, and the sine (yes, sine, not sign) of the angle must also be equal to that same value. But then, $1^2+1^2=1+1=2$ and $(-1)^2+(-1)^2=1+1=2$ both contradict the Pythagorean trigonometric identity.

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