Question

Prove the following trigonometric identities:
(1+cotA-cosecA)(1+tanA+secA)=2

Answer 1

Step-by-step explanation:

• We know that

        $\mathbf{\cot A=\frac{\cos A}{\sin A}}$

        $\mathbf{\tan A=\frac{\sin A}{\cos A}}$

        $\mathbf{\csc A=\frac{1}{\sin A}}$

        $\mathbf{\sec A=\frac{1}{\cos A}}$

        $\mathbf{\sin^{2} A+\cos^{2} A=1 }$                   ...1)

        $\mathbf{X^{2}-Y^{2}=(X+Y)(X-Y)}$     ...2)

        $\mathbf{(X+Y)^{2}=X^{2}+Y^{2}+2XY}$    ...3)

• From L.H.S

        $\mathbf{(1+\cot A-\csc A)(1+\tan A+\sec A)}$

        $\mathbf{\left ( 1+\frac{\cos A}{\sin A}-\frac{1}{\sin A} \right )\left ( 1+\frac{\sin A}{\cos A}+\frac{1}{\cos A} \right )}$

        $\mathbf{\frac{(\sin A+\cos A-1)}{\cos A}\frac{(\sin A+\cos A+1)}{\sin A}}$  

• By using identity of equation 2)

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

• By using identity of equation 3)

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

• By using equation 1)

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

       $\mathbf{\frac{2\sin A \cos A}{\sin A \cos A}}$

       2 = R.H.S