Question

Prove the following trigonometric identities:
(tanθ+1/cosθ)²+(tanθ-1/cosθ)²=2(1+sin²θ/1-sin²θ)

Answer 1

Step-by-step explanation:

• Consider L.H.S

       $\mathbf{\left ( \tan \Theta +\frac{1}{\cos \Theta } \right )^{2}+\left ( \tan \Theta -\frac{1}{\cos \Theta } \right )^{2}}$

       $\mathbf{\left ( \frac{\tan \Theta\times \cos \Theta+1}{\cos \Theta} \right )^{2}+\left ( \frac{\tan \Theta\times \cos \Theta-1}{\cos \Theta} \right )^{2}}$    (where $\mathbf{\tan \Theta \cos \Theta =\sin \Theta }$ )  

       $\mathbf{\left ( \frac{\sin \Theta +1}{\cos \Theta} \right )^{2}+\left ( \frac{\sin \Theta -1}{\cos \Theta} \right )^{2}}$

       $\mathbf{\frac{(\sin \Theta +1)^{2}}{\cos^{2} \Theta }+\frac{(\sin \Theta -1)^{2}}{\cos^{2} \Theta}}$

       $\mathbf{\frac{(\sin \Theta +1)^{2}+(\sin \Theta -1)^{2}}{\cos^{2} \Theta}}$

       $\mathbf{\frac{1+\sin^{2}\Theta +2\sin \Theta +1+\sin^{2}\Theta -2\sin \Theta}{\cos^{2} \Theta}}$

       $\mathbf{\frac{2+2\sin^{2}\Theta }{\cos^{2} \Theta}}$

       $\mathbf{\frac{2(1+\sin^{2}\Theta)}{\cos^{2} \Theta}}$      (where $\mathbf{\cos^{2} \Theta =1-\sin^{2}\Theta}$ )  

       $\mathbf{\frac{2(1+\sin^{2}\Theta)}{1-\sin^{2}\Theta}}$      = R.H.S