Question

Prove that: $\csc (\pi /4+\theta )\csc (\pi /4-\theta )=\sec (\pi /4+\theta )\sec (\pi /4-\theta )=2\sec 2\theta$

Answer 1

LHS $=\frac{1}{\sin ({45}^o+\theta )\sin ({45}^o-\theta )}=\frac{1}{{\sin }^2{45}^o-{\sin }^2\theta }=\frac{2}{1-2{\sin }^2\theta }=\frac{2}{\cos 2\theta }=2\sec 2\theta$
Similarly, $\sec (\frac{\pi }{4}+\theta )\sec (\frac{\pi }{4}-\theta )=\frac{1}{{\cos }^2{45}^o-{\sin }^2{45}^o}=2\sec 2\theta$