Answer:
Let [tex]\left(1+\frac{1}{\tan^{2}A} \right)\cdot \left(1+\frac{1}{\cot^{2}A} \right)[/tex], we proceed to prove the trigonometric expression by trigonometric identity:
1) [tex]\left(1+\frac{1}{\tan^{2}A} \right)\cdot \left(1+\frac{1}{\cot^{2}A} \right)[/tex] Given
2) [tex]\left(1+\frac{\cos^{2}A}{\sin^{2}A} \right)\cdot \left(1+\frac{\sin^{2}A}{\cos^{2}A} \right)[/tex] [tex]\tan A = \frac{1}{\cot A} = \frac{\sin A}{\cos A}[/tex]
3) [tex]\left(\frac{\sin^{2}A+\cos^{2}A}{\sin^{2}A} \right)\cdot \left(\frac{\cos^{2}A+\sin^{2}A}{\cos^{2}A} \right)[/tex]
4) [tex]\left(\frac{1}{\sin^{2}A} \right)\cdot \left(\frac{1}{\cos^{2}A} \right)[/tex] [tex]\sin^{2}A+\cos^{2}A = 1[/tex]
5) [tex]\frac{1}{\sin^{2}A\cdot \cos^{2}A}[/tex]
6) [tex]\frac{1}{\sin^{2}A\cdot (1-\sin^{2}A)}[/tex] [tex]\sin^{2}A+\cos^{2}A = 1[/tex]
7) [tex]\frac{1}{\sin^{2}A-\sin^{4}A}[/tex] Result
Step-by-step explanation:
Let [tex]\left(1+\frac{1}{\tan^{2}A} \right)\cdot \left(1+\frac{1}{\cot^{2}A} \right)[/tex], we proceed to prove the trigonometric expression by trigonometric identity:
1) [tex]\left(1+\frac{1}{\tan^{2}A} \right)\cdot \left(1+\frac{1}{\cot^{2}A} \right)[/tex] Given
2) [tex]\left(1+\frac{\cos^{2}A}{\sin^{2}A} \right)\cdot \left(1+\frac{\sin^{2}A}{\cos^{2}A} \right)[/tex] [tex]\tan A = \frac{1}{\cot A} = \frac{\sin A}{\cos A}[/tex]
3) [tex]\left(\frac{\sin^{2}A+\cos^{2}A}{\sin^{2}A} \right)\cdot \left(\frac{\cos^{2}A+\sin^{2}A}{\cos^{2}A} \right)[/tex]
4) [tex]\left(\frac{1}{\sin^{2}A} \right)\cdot \left(\frac{1}{\cos^{2}A} \right)[/tex] [tex]\sin^{2}A+\cos^{2}A = 1[/tex]
5) [tex]\frac{1}{\sin^{2}A\cdot \cos^{2}A}[/tex]
6) [tex]\frac{1}{\sin^{2}A\cdot (1-\sin^{2}A)}[/tex] [tex]\sin^{2}A+\cos^{2}A = 1[/tex]
7) [tex]\frac{1}{\sin^{2}A-\sin^{4}A}[/tex] Result
