Prove $\sec^2 A + \csc^2 A = 4 \csc^2 2A$

Question

Hi would someone kindly help show me how to prove

$$\sec^2 A + \csc^2 A = 4 \csc^2 2A$$

I got...

\begin{align}\sec^2 A + \csc^2 A &= 1/\cos^2 A + 1/\sin^2 A = (\sin^2 A + \cos^2 A) / [(\cos^2 A)(\sin^2 A)]\\ &= 1 / [(\cos^2 A)(\sin^2 A)]\end{align}

but I am not sure on what algebraic or trigonometric trick is required to show RHS.

Thanks!

Answer 1

$$\sec^2 A + \csc^2 A = \frac {1}{\cos^2 A} + \frac {1}{\sin^2 A}= \frac {\sin^2 A +\cos^2 A}{\sin^2 A \cos^2 A}$$

Note that $$\sin A \cos A = (1/2)\sin 2A $$

Therefore $$\frac {\sin^2 A +\cos^2 A}{\sin^2 A \cos^2 A}=\frac {1}{\sin^2 A \cos^2 A} = 4\csc^2 2A$$

Answer 2

$$\frac1{\cos^2A}+\frac1{\sin^2A}=\frac1{\cos^2A\sin^2A}$$ Next use the duplication formula for the sine.