What is the value of $$\tan \left(\frac{\pi}{4} + A\right) \times \tan \left(\frac{3\pi}{4} + A\right)?$$
We know $$\tan\ \frac{\pi}{4}=1\ ;\ \tan\ \frac{3}{4}\pi\ \ =-1\ $$
and tan (A+B) = (tan A +tan B )/1-tan A tan B
Now using formula we get $$\tan\left(\frac{\pi}{4}+A\ \right)\times\ \tan\left(\frac{3}{4}\pi\ +A\right)$$
=$$\frac{1+\tan\ A}{1-\tan\ A}\times\ \frac{-1+\tan\ A}{1+\tan\ A}$$
=-1
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