# Are there any angles for which cos and cosec give same values?

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 $$\cos Q=\csc Q$$

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.
• +1. Unless $Q$ is complex valued – Qurultay Feb 14 at 18:53
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.