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Q2(ii):
Choose the correct option and justify your choice : (ii) $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =$
Choose the correct option and justify your choice : (ii) $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =$
Solution :
Given: The trigonometric expression $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ}$.
To Find: The numerical value of the expression and identify the correct option among the standard trigonometric values.
Step 1: Recall the trigonometric ratio for $45^\circ$
From the standard trigonometric table for specific angles, we know that:
$\tan 45^\circ = 1$
Step 2: Substitute the value into the expression
The given expression is:
$E = \frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ}$
Substituting $\tan 45^\circ = 1$ into the expression:
$E = \frac{1 - (1)^2}{1 + (1)^2}$
Step 3: Perform arithmetic simplification
Calculate the square of the value:
$(1)^2 = 1 \times 1 = 1$
Substitute this back into the fraction:
$E = \frac{1 - 1}{1 + 1}$
Perform the subtraction in the numerator and the addition in the denominator:
$E = \frac{0}{2}$
Step 4: Final evaluation
Any fraction with a numerator of $0$ and a non-zero denominator is equal to $0$.
$E = 0$
Step 5: Justification and Comparison with Options
We evaluate the standard trigonometric values typically provided in such multiple-choice questions:
(A) $\tan 90^\circ$ (Undefined)
(B) $1$
(C) $\sin 45^\circ = \frac{1}{\sqrt{2}}$
(D) $0$
Since our calculated value is $0$, the expression is equal to $0$.
Final Answer: The value of the expression is 0.
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.2
- Q1(i): Evaluate the following : (i) $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$
- Q1(ii): Evaluate the following : (ii) $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$
- Q1(iii): Evaluate the following : (iii) $\frac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}$
- Q1(iv): Evaluate the following : (iv) $\frac{\sin 30^\circ + \tan 45^\circ – \text{cosec } 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$
- Q1(v): Evaluate the following : (v) $\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
- Q2(i): Choose the correct option and justify your choice : (i) $\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =$
- Q2(iii): Choose the correct option and justify your choice : (iii) $\sin 2A = 2 \sin A$ is true when $A =$
- Q2(iv): Choose the correct option and justify your choice : (iv) $\frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =$
- Q3: If $\tan (A + B) = \sqrt{3}$ and $\tan (A – B) = \frac{1}{\sqrt{3}}$; $0^\circ < A + B \le 90^\circ$; $A > B$, find $A$ and $B$.
- Q4(i): State whether the following are true or false. Justify your answer. (i) $\sin (A + B) = \sin A + \sin B$.
- Q4(ii): State whether the following are true or false. Justify your answer. (ii) The value of $\sin \theta$ increases as $\theta$ increases.
- Q4(iii): State whether the following are true or false. Justify your answer. (iii) The value of $\cos \theta$ increases as $\theta$ increases.
- Q4(iv): State whether the following are true or false. Justify your answer. (iv) $\sin \theta = \cos \theta$ for all values of $\theta$.
- Q4(v): State whether the following are true or false. Justify your answer. (v) $\cot A$ is not defined for $A = 0^\circ$.
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
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