Evaluate $$\sec\big(\frac{\pi}{4}+x\big).\sec\big(\frac{\pi}{4}-x\big)$$
This can be evaluated as $$ \sec\big(\frac{\pi}{4}+x\big).\sec\big(\frac{\pi}{4}-x\big)=\frac{1}{\cos\big(\frac{\pi}{4}+x\big).\cos\big(\frac{\pi}{4}+x\big)}\\ =\frac{1}{\cos^2\frac{\pi}{4}-\sin^2x}=\frac{2}{1-2\sin^2x}=2\sec2x $$ But, when I try the following by making use of the formula $2\cos x\cos y=\cos(x+y)+\cos(x-y)$, $$ \sec\big(\frac{\pi}{4}+x\big).\sec\big(\frac{\pi}{4}-x\big)=\frac{2}{2\cos\big(\frac{\pi}{4}+x\big).\cos\big(\frac{\pi}{4}+x\big)}\\ =\frac{2}{\cos\frac{\pi}{2}+\cos2x}=\frac{2}{\text{not defined} +\cos2x} $$
Why do I get such a problem in my second attempt ?
why will cos $\pi/2$ not be defined?