Chứng minh rằng:
a) \(\sin \alpha + \cos \alpha = \sqrt 2 \sin (\alpha + {\pi \over 4})\)
b) \(\sin \alpha - \cos \alpha = \sqrt 2 \sin (\alpha - {\pi \over 4})\)
c) \(\tan ({\pi \over 4} - \alpha ) = {{1 - \tan \alpha } \over {1 + \tan \alpha }}\,\,(\alpha \ne {\pi \over 2} + k\pi ;\,\,\alpha \ne {{3\pi } \over 4} + k\pi )\)
d) \(\tan ({\pi \over 4} + \alpha ) = {{1 + \tan \alpha } \over {1 - \tan \alpha }}\,\,(\alpha \ne {\pi \over 2} + k\pi ;\,\,\alpha \ne {\pi \over 4} + k\pi )\)