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Q1(ii):
Evaluate the following : (ii) $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$

Solution :

Given: An algebraic expression involving trigonometric ratios: $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$.

To Find: The numerical value of the given expression.

Step 1: Identification of Trigonometric Values
To evaluate the expression, we must recall the standard trigonometric ratios for the given angles from the trigonometric table:

  • $\tan 45^\circ = 1$
  • $\cos 30^\circ = \frac{\sqrt{3}}{2}$
  • $\sin 60^\circ = \frac{\sqrt{3}}{2}$

Step 2: Substitution of Values into the Expression
Substitute the identified values into the expression $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$:

$= 2(1)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{\sqrt{3}}{2}\right)^2$

Step 3: Performing Arithmetic Operations
Now, we simplify each term step-by-step:

First, calculate the squares:

  • $(1)^2 = 1$
  • $\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{(\sqrt{3})^2}{(2)^2} = \frac{3}{4}$

Substitute these back into the expression:

$= 2(1) + \frac{3}{4} - \frac{3}{4}$

Step 4: Final Simplification
Perform the multiplication and addition/subtraction:

$= 2 + \frac{3}{4} - \frac{3}{4}$

[Since $\frac{3}{4} - \frac{3}{4} = 0$]

$= 2 + 0$

$= 2$

Final Answer: 2


More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.2


CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry


Chapters in CBSE - Class 10 Mathematics


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