Q4(viii):

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (viii) $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$

Given: An identity involving trigonometric functions of an acute angle $A$: $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2$.

To Prove: $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$.

Step 1: Expanding the Left Hand Side (LHS)

We start with the expression: $LHS = (\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2$.

Using the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$, we expand both terms:

$(\sin A + \text{cosec } A)^2 = \sin^2 A + 2\sin A \cdot \text{cosec } A + \text{cosec}^2 A$

$(\cos A + \sec A)^2 = \cos^2 A + 2\cos A \cdot \sec A + \sec^2 A$

Step 2: Simplifying the expanded terms

Recall the reciprocal identities: $\text{cosec } A = \frac{1}{\sin A}$ and $\sec A = \frac{1}{\cos A}$.

Substituting these into the middle terms:

$2\sin A \cdot \text{cosec } A = 2\sin A \cdot \left(\frac{1}{\sin A}\right) = 2$

$2\cos A \cdot \sec A = 2\cos A \cdot \left(\frac{1}{\cos A}\right) = 2$

Now, substitute these back into the expanded expression:

$LHS = \sin^2 A + 2 + \text{cosec}^2 A + \cos^2 A + 2 + \sec^2 A$

Step 3: Grouping terms using Trigonometric Identities

Rearrange the terms to group $\sin^2 A$ and $\cos^2 A$ together:

$LHS = (\sin^2 A + \cos^2 A) + 2 + 2 + \text{cosec}^2 A + \sec^2 A$

Using the Pythagorean identity $\sin^2 A + \cos^2 A = 1$:

$LHS = 1 + 4 + \text{cosec}^2 A + \sec^2 A$

$LHS = 5 + \text{cosec}^2 A + \sec^2 A$

Step 4: Converting to $\tan^2 A$ and $\cot^2 A$

We use the trigonometric identities: $\text{cosec}^2 A = 1 + \cot^2 A$ and $\sec^2 A = 1 + \tan^2 A$.

Substitute these into the expression:

$LHS = 5 + (1 + \cot^2 A) + (1 + \tan^2 A)$

$LHS = 5 + 1 + 1 + \tan^2 A + \cot^2 A$

$LHS = 7 + \tan^2 A + \cot^2 A$

Step 5: Conclusion

Since the simplified LHS is equal to the Right Hand Side (RHS):

$LHS = RHS = 7 + \tan^2 A + \cot^2 A$

Final Answer: The identity $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$ is proven.