Formula used:tan(A + B) = \(\displaystyle \frac{tan\ A+\ tan\ B}{1\ -\ tan\ A.\ tan\ B}\)sin 2? = 2 sin ? cos ? Calculation:tan (45° + ?) = 1 + sin 2?? \(\displaystyle \frac{tan\ 45^o+\ tan\ ?}{1\ -\ tan\ 45^o.\ tan\ ?}\) = 1 + sin 2?? \(\displaystyle \frac{1+\ tan\ ?}{1\ -\ tan\ ?}\) = 1 + sin 2?Since, tan ? = sin ?/cos ? ? \(\displaystyle \frac{cos\ ?+\ sin\ ?}{cos\ ?-\ sin\ ?}\) = 1 + sin 2?? cos ? + sin ? = cos ? - sin ? + sin 2?(cos ? - sin ?)? 2 sin ? = sin 2?(cos ? - sin ?)We know that, sin 2? = 2 sin ? cos ? ? 2 sin ? = 2 sin ?. cos ? (cos ? - sin ?)? 1 = cos ? (cos ? - sin ?)? 1 = cos2? - cos ?.sin ?? sin2? = - cos ?.sin ?? sin2? + cos ?.sin ? = 0? sin ?(sin ? + cos ?) = 0? sin ? = 0 or sin ? = - cos ?? sin ? = 0 or tan ? = -1? ? = 0 or ? = \(\displaystyle -\frac{\pi}{4}\)As ? lies in \(\displaystyle -\frac{\pi}{4}< ? <\frac{\pi}{4},\) ? ? = 0ocos 2? = cos (2 × 0) = 1? cos 2? = 1