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Q3(iii):
Choose the correct option. Justify your choice. (iii) $(\sec A + \tan A) (1 – \sin A) =$
Choose the correct option. Justify your choice. (iii) $(\sec A + \tan A) (1 – \sin A) =$
Solution :
Given: An expression $(\sec A + \tan A)(1 - \sin A)$.
To find: The simplified value of the given expression by choosing the correct option among the standard trigonometric identities.
Step 1: Expressing trigonometric ratios in terms of sine and cosine.
We know the fundamental definitions of trigonometric ratios:
$\sec A = \frac{1}{\cos A}$ [Since secant is the reciprocal of cosine]
$\tan A = \frac{\sin A}{\cos A}$ [Since tangent is the ratio of sine to cosine]
Step 2: Substituting these values into the given expression.
Let the expression be $E$.
$E = \left( \frac{1}{\cos A} + \frac{\sin A}{\cos A} \right) (1 - \sin A)$
Step 3: Simplifying the term inside the parentheses.
Since the denominators are the same, we can combine the fractions:
$E = \left( \frac{1 + \sin A}{\cos A} \right) (1 - \sin A)$
Step 4: Performing the multiplication.
Multiply the numerators together:
$E = \frac{(1 + \sin A)(1 - \sin A)}{\cos A}$
Step 5: Applying the algebraic identity $(a + b)(a - b) = a^2 - b^2$.
Here, $a = 1$ and $b = \sin A$.
$E = \frac{1^2 - \sin^2 A}{\cos A}$
$E = \frac{1 - \sin^2 A}{\cos A}$
Step 6: Applying the Pythagorean identity.
We know the identity: $\sin^2 A + \cos^2 A = 1$.
Rearranging this gives: $1 - \sin^2 A = \cos^2 A$.
Substituting this into our expression:
$E = \frac{\cos^2 A}{\cos A}$
Step 7: Final simplification.
$E = \frac{\cos A \cdot \cos A}{\cos A}$
$E = \cos A$ [By canceling the common factor $\cos A$ in the numerator and denominator]
Final Answer: The simplified value of the expression is $\cos A$.
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(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) $(\text{cosec } \theta โ \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$
- 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) $\frac{1 + \sec A}{\sec A} = \frac{\sin^2 A}{1 โ \cos A}$ [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) $(\text{cosec } A โ \sin A) (\sec A โ \cos A) = \frac{1}{\tan A + \cot A}$ [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) $\frac{\cos A โ \sin A + 1}{\cos A + \sin A โ 1} = \text{cosec } A + \cot A$, 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) $\sqrt{\frac{1 + \sin A}{1 โ \sin A}} = \sec A + \tan A$
- 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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