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Q4(viii):
Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (viii) $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$
Solution :
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.
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.3
- Q1: Express the trigonometric ratios $\sin A$, $\sec A$ and $\tan A$ in terms of $\cot A$.
- Q2: Write all the other trigonometric ratios of $\angle A$ in terms of $\sec A$.
- Q3(i): Choose the correct option. Justify your choice. (i) $9 \sec^2 A – 9 \tan^2 A =$
- Q3(ii): Choose the correct option. Justify your choice. (ii) $(1 + \tan \theta + \sec \theta) (1 + \cot \theta – \text{cosec } \theta) =$
- Q3(iii): Choose the correct option. Justify your choice. (iii) $(\sec A + \tan A) (1 – \sin A) =$
- Q3(iv): Choose the correct option. Justify your choice. (iv) $\frac{1 + \tan^2 A}{1 + \cot^2 A} =$
- Q4(i): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) $(\text{cosec } \theta – \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$
- Q4(ii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ii) $\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 \sec A$
- Q4(iii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iii) $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \text{cosec } \theta$ [Hint : Write the expression in terms of $\sin \theta$ and $\cos \theta$]
- Q4(iv): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iv) $\frac{1 + \sec A}{\sec A} = \frac{\sin^2 A}{1 – \cos A}$ [Hint : Simplify LHS and RHS separately]
- Q4(ix): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ix) $(\text{cosec } A – \sin A) (\sec A – \cos A) = \frac{1}{\tan A + \cot A}$ [Hint : Simplify LHS and RHS separately]
- Q4(v): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (v) $\frac{\cos A – \sin A + 1}{\cos A + \sin A – 1} = \text{cosec } A + \cot A$, using the identity $\text{cosec}^2 A = 1 + \cot^2 A$.
- Q4(vi): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vi) $\sqrt{\frac{1 + \sin A}{1 – \sin A}} = \sec A + \tan A$
- Q4(vii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vii) $\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$
- Q4(x): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (x) $(\frac{1 + \tan^2 A}{1 + \cot^2 A}) = (\frac{1 - \tan A}{1 - \cot A})^2 = \tan^2 A$
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