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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]
Solution :
Given: The trigonometric identity $(\text{cosec } A - \sin A)(\sec A - \cos A) = \frac{1}{\tan A + \cot A}$, where $A$ is an acute angle.
To Prove: The Left Hand Side (LHS) is equal to the Right Hand Side (RHS).
Step 1: Simplifying the Left Hand Side (LHS)
LHS = $(\text{cosec } A - \sin A)(\sec A - \cos A)$
Using the reciprocal identities $\text{cosec } A = \frac{1}{\sin A}$ and $\sec A = \frac{1}{\cos A}$:
LHS = $\left( \frac{1}{\sin A} - \sin A \right) \left( \frac{1}{\cos A} - \cos A \right)$
Taking the common denominator for each bracket:
LHS = $\left( \frac{1 - \sin^2 A}{\sin A} \right) \left( \frac{1 - \cos^2 A}{\cos A} \right)$
Applying the Pythagorean identity $\sin^2 A + \cos^2 A = 1$, which implies $1 - \sin^2 A = \cos^2 A$ and $1 - \cos^2 A = \sin^2 A$:
LHS = $\left( \frac{\cos^2 A}{\sin A} \right) \left( \frac{\sin^2 A}{\cos A} \right)$
Canceling the common terms in the numerator and denominator:
LHS = $\cos A \cdot \sin A$
Step 2: Simplifying the Right Hand Side (RHS)
RHS = $\frac{1}{\tan A + \cot A}$
Using the quotient identities $\tan A = \frac{\sin A}{\cos A}$ and $\cot A = \frac{\cos A}{\sin A}$:
RHS = $\frac{1}{\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}}$
Finding the common denominator in the denominator of the fraction:
RHS = $\frac{1}{\frac{\sin^2 A + \cos^2 A}{\cos A \sin A}}$
Applying the Pythagorean identity $\sin^2 A + \cos^2 A = 1$:
RHS = $\frac{1}{\frac{1}{\cos A \sin A}}$
By the property of reciprocals of fractions ($\frac{1}{1/x} = x$):
RHS = $\sin A \cos A$
Step 3: Conclusion
Since LHS = $\sin A \cos A$ and RHS = $\sin A \cos A$, we have shown that LHS = RHS.
Final Answer: Hence, it is proved that $(\text{cosec } A - \sin A)(\sec A - \cos A) = \frac{1}{\tan A + \cot A}$.
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(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(viii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (viii) $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$
- 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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