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Q3(ii):
Choose the correct option. Justify your choice. (ii) $(1 + \tan \theta + \sec \theta) (1 + \cot \theta – \text{cosec } \theta) =$
Choose the correct option. Justify your choice. (ii) $(1 + \tan \theta + \sec \theta) (1 + \cot \theta – \text{cosec } \theta) =$
Solution :
Given: The trigonometric expression $(1 + \tan \theta + \sec \theta)(1 + \cot \theta - \text{cosec } \theta)$.
To Find: The simplified value of the given expression.
Step 1: Expressing all trigonometric ratios in terms of sine and cosine.
We use the following fundamental trigonometric identities:
$\tan \theta = \frac{\sin \theta}{\cos \theta}$
$\sec \theta = \frac{1}{\cos \theta}$
$\cot \theta = \frac{\cos \theta}{\sin \theta}$
$\text{cosec } \theta = \frac{1}{\sin \theta}$
Substituting these into the expression:
Expression $= \left(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta}\right) \left(1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta}\right)$
Step 2: Simplifying the terms inside each bracket.
For the first bracket, find a common denominator ($\cos \theta$):
$\left(\frac{\cos \theta + \sin \theta + 1}{\cos \theta}\right)$
For the second bracket, find a common denominator ($\sin \theta$):
$\left(\frac{\sin \theta + \cos \theta - 1}{\sin \theta}\right)$
Step 3: Multiplying the two fractions.
Expression $= \frac{(\sin \theta + \cos \theta + 1)(\sin \theta + \cos \theta - 1)}{\sin \theta \cos \theta}$
Step 4: Applying the algebraic identity $(a + b)(a - b) = a^2 - b^2$.
Let $a = (\sin \theta + \cos \theta)$ and $b = 1$.
Numerator $= (\sin \theta + \cos \theta)^2 - (1)^2$
Expanding $(\sin \theta + \cos \theta)^2$ using $(a + b)^2 = a^2 + b^2 + 2ab$:
Numerator $= (\sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta) - 1$
Step 5: Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$.
Numerator $= (1 + 2 \sin \theta \cos \theta) - 1$
Numerator $= 2 \sin \theta \cos \theta$
Step 6: Final simplification.
Expression $= \frac{2 \sin \theta \cos \theta}{\sin \theta \cos \theta}$
Canceling the common terms $\sin \theta \cos \theta$ (assuming $\sin \theta \neq 0$ and $\cos \theta \neq 0$):
Expression $= 2$
Final Answer: 2
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(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(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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