Question

prove that sec^4a(1-sin^4a) - 2tan^2a = 1

Answer 1

Prove:

$$\begin{array}{l} \sec^{4} a \cdot (1 - \sin^{4} a) - 2 \tan^{2} a = 1 \\ \sec^{4} a \cdot (1 - \sin^{2} a) \cdot (1 + \sin^{2} a) - 2 \tan^{2} a = \\ = \frac{1}{\cos^{4} a} \cdot \cos^{2} a \cdot (1 + \sin^{2} a) - 2 \tan^{2} a = \\ = \frac{1}{\cos^{2} a} \cdot (1 + \sin^{2} a) - \frac{2 \sin^{2} a}{\cos^{2} a} = \\ = \frac{1}{\cos^{2} a} + \frac{\sin^{2} a}{\cos^{2} a} - \frac{2 \sin^{2} a}{\cos^{2} a} = \\ = \frac{1}{\cos^{2} a} - \frac{\sin^{2} a}{\cos^{2} a} = \\ = \frac{(1 - \sin^{2} a)}{\cos^{2} a} = \\ = \frac{\cos^{2} a}{\cos^{2} a} = 1 \Rightarrow \text{Proved}. \end{array}$$