$e^{i\theta}= \cos \theta + i \sin \theta , $ $e^{i7 \theta}= (\cos \theta + i \sin \theta)^7 $ $\cos 7\theta + i \sin 7 \theta $ $= \cos^7 \theta + ^7C_1 \cos^6 \theta (i \sin \theta)^1 +^7C_2 \cos^5 \theta (i \sin \theta)^2 +^7 C_3 \cos^4 \theta ( i \sin \theta)^3 + ^7C_4 \cos^3 \theta (i \sin \theta)^4+ ^7C_5 \cos^2 (i \sin \theta)^5 +^{7}C_6 \cos^1 \theta (i \sin \theta)^6 + (i \sin \theta)^7 $ $=\cos^7\theta + 7i \cos^6 \theta \sin \theta - 21 \cos^5 \theta \sin^2 \theta - 35 i \cos^4 \theta \sin^3 \theta +35 \cos^3 \theta \sin^4 \theta + 21 i \cos^2 \theta \sin^5 \theta -7 \cos \theta \sin^6 \theta - i \sin^7 \theta $ ${ i^2 = -1, \ i^3=-i, \ i^4=1, \ i^5=i , \ i^6=-1, \ i^7=-i}$ {7th row of Pascal Triangle is 1,7,21,35,35,21,7,1} $ \cos 7\theta + i \sin 7 \theta = (\cos^7 \theta -21 \cos^5 \theta \sin^2 \theta +35 \cos^3 \theta \sin^4 \theta - 7 \cos \theta \sin^6 \theta)+ i (7 \cos^6 \theta \sin \theta -35 \cos^4 \theta + \sin^3 \theta +21 \cos^2 \theta \sin^5 \theta -\sin^7 \theta)$ Comparing real and Imaginary parts, $\cos 7 \theta = \cos^7 \theta -21 \cos^5 \theta \sin^2 \theta +35 \cos^3 \theta \sin^4 \theta - 7 \cos \theta \sin^6 \theta$ $\sin 7 \theta = 7 \cos^6 \theta \sin \theta -35 \cos^4 \theta \sin^3 \theta + 21 \cos^2 \theta \sin^5 \theta -\sin^7 \theta$ $\tan 7 \theta = \dfrac {\sin 7 \theta}{\cos 7 \theta} = \dfrac {7 \cos^6 \theta \sin \theta - 35 \cos^4 \theta \sin^3 \theta + 21 \cos^2 \sin^5 \theta - \sin^7 \theta} {\cos^7 \theta -21 \cos^5 \theta \sin^2 \theta +35 \cos^3 \theta \sin^4 \theta -7 \cos \theta \sin^6 \theta }$ Divide numerator and denominator by $\cos^7 \theta$ $\tan 7 \theta = \dfrac {\frac {7 \cos^6 \theta \sin \theta}{\cos^7 \theta} -35 \frac {\cos^4 \theta \sin^3 \theta}{\cos^7 \theta} + \frac {21 \cos^2 \theta \sin^5 \theta}{\cos^7 \theta} - \frac {\sin^7 \theta}{\cos^7 \theta}}{ 1- \frac {21 \cos^5 \theta \sin^2 \theta}{\cos^7 \theta} + 35 \frac {\cos^3 \theta \sin \theta}{\cos^7 \theta}- \frac {7\cos \theta \sin^6 \theta}{\cos^7 \theta}}$ $\tan 7 \theta = \dfrac {7 \tan \theta -35 \tan^3 \theta +21 \tan^5 \theta - \tan^7 \theta}{1-21 \tan^2 +35 \tan^4 \theta -7\tan^6 \theta}$

Hence Proved.