Question

Solve the above que no. 55

Answer 1

Answer:

Let $\left(1+\frac{1}{\tan^{2}A} \right)\cdot \left(1+\frac{1}{\cot^{2}A} \right)$, we proceed to prove the trigonometric expression by trigonometric identity:

1) $\left(1+\frac{1}{\tan^{2}A} \right)\cdot \left(1+\frac{1}{\cot^{2}A} \right)$ Given

2) $\left(1+\frac{\cos^{2}A}{\sin^{2}A} \right)\cdot \left(1+\frac{\sin^{2}A}{\cos^{2}A} \right)$   $\tan A = \frac{1}{\cot A} = \frac{\sin A}{\cos A}$

3) $\left(\frac{\sin^{2}A+\cos^{2}A}{\sin^{2}A} \right)\cdot \left(\frac{\cos^{2}A+\sin^{2}A}{\cos^{2}A} \right)$    

4) $\left(\frac{1}{\sin^{2}A} \right)\cdot \left(\frac{1}{\cos^{2}A} \right)$    $\sin^{2}A+\cos^{2}A = 1$

5) $\frac{1}{\sin^{2}A\cdot \cos^{2}A}$

6) $\frac{1}{\sin^{2}A\cdot (1-\sin^{2}A)}$    $\sin^{2}A+\cos^{2}A = 1$

7) $\frac{1}{\sin^{2}A-\sin^{4}A}$ Result

Step-by-step explanation:

Let $\left(1+\frac{1}{\tan^{2}A} \right)\cdot \left(1+\frac{1}{\cot^{2}A} \right)$, we proceed to prove the trigonometric expression by trigonometric identity:

1) $\left(1+\frac{1}{\tan^{2}A} \right)\cdot \left(1+\frac{1}{\cot^{2}A} \right)$ Given

2) $\left(1+\frac{\cos^{2}A}{\sin^{2}A} \right)\cdot \left(1+\frac{\sin^{2}A}{\cos^{2}A} \right)$   $\tan A = \frac{1}{\cot A} = \frac{\sin A}{\cos A}$

3) $\left(\frac{\sin^{2}A+\cos^{2}A}{\sin^{2}A} \right)\cdot \left(\frac{\cos^{2}A+\sin^{2}A}{\cos^{2}A} \right)$    

4) $\left(\frac{1}{\sin^{2}A} \right)\cdot \left(\frac{1}{\cos^{2}A} \right)$    $\sin^{2}A+\cos^{2}A = 1$

5) $\frac{1}{\sin^{2}A\cdot \cos^{2}A}$

6) $\frac{1}{\sin^{2}A\cdot (1-\sin^{2}A)}$    $\sin^{2}A+\cos^{2}A = 1$

7) $\frac{1}{\sin^{2}A-\sin^{4}A}$ Result