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

Question 7

Step 1: Recall the trigonometric identity: \(\sin\theta = \cos(90^\circ - \theta)\).

Step 2: In the question, we are given \(\cos 9\alpha = \sin \alpha\). Using the identity, replace \(\sin\alpha\) by \(\cos(90^\circ - \alpha)\).

So, \(\cos 9\alpha = \cos(90^\circ - \alpha)\).

Step 3: If two cos values are equal, their angles must also be equal (as long as both lie in the given range). Hence, \(9\alpha = 90^\circ - \alpha\).

Step 4: Solve for \(\alpha\):

\(9\alpha + \alpha = 90^\circ\)

\(10\alpha = 90^\circ\)

\(\alpha = 9^\circ\)

Step 5: Now we need \(\tan 5\alpha\).

Substitute \(\alpha = 9^\circ\):

\(\tan 5\alpha = \tan(5 \times 9^\circ) = \tan 45^\circ\).

Step 6: We know \(\tan 45^\circ = 1\).

Final Answer: \(\tan 5\alpha = 1\).