NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 8: Introduction to Trignometry and Its Applications - Exercise 8.1

Question 13

Step 1: We are given that \(\sin\theta - \cos\theta = 0\).

This means \(\sin\theta = \cos\theta\).

Step 2: From trigonometry, we know that:

\(\sin^2\theta + \cos^2\theta = 1\) (Pythagoras identity).

Step 3: Since \(\sin\theta = \cos\theta\), let each of them be equal to \(x\).

So, \(\sin\theta = x\) and \(\cos\theta = x\).

Step 4: Substitute into the identity:

\(x^2 + x^2 = 1\)

\(2x^2 = 1\)

\(x^2 = \dfrac{1}{2}\)

\(x = \dfrac{1}{\sqrt{2}}\)

Step 5: Now calculate the required expression:

\(\sin^4\theta + \cos^4\theta = x^4 + x^4 = 2x^4\)

Step 6: Since \(x = \dfrac{1}{\sqrt{2}}\):

\(x^4 = \left(\dfrac{1}{\sqrt{2}}\right)^4 = \dfrac{1}{4}\)

So, \(2x^4 = 2 \times \dfrac{1}{4} = \dfrac{1}{2}\).

Final Answer: \(\sin^4\theta + \cos^4\theta = \dfrac{1}{2}\).