Advertisement Remove all ads

Prove the Following Identities Tan 3 X 1 + Tan 2 X + Cot 3 X 1 + Cot 2 X = 1 − 2 Sin 2 X Cos 2 X Sin X Cos X - Mathematics

Prove the following identities

\[\frac{\tan^3 x}{1 + \tan^2 x} + \frac{\cot^3 x}{1 + \cot^2 x} = \frac{1 - 2 \sin^2 x \cos^2 x}{\sin x \cos x}\]
Advertisement Remove all ads

Solution

\[\text{ LHS }= \frac{\tan^3 x}{1 + \tan^2 x} + \frac{\cot^3 x}{1 + \cot^2 x}\]
\[ = \frac{\tan^3 x}{\sec^2 x} + \frac{\cot^3 x}{{cosec}^2 x}\]
\[ = \frac{\frac{\sin^3 x}{\cos^3 x}}{\frac{1}{\cos^2 x}} + \frac{\frac{\cos^3 x}{\sin^3 x}}{\frac{1}{\sin^2 x}}\]
\[ = \frac{\sin^3 x}{\cos^3 x} \times \frac{\cos^2 x}{1} + \frac{\cos^3 x}{\sin^3 x} \times \frac{\sin^2 x}{1}\]
\[ = \frac{\sin^3 x}{\cos x} + \frac{\cos^3 x}{\sin x}\]
\[ = \frac{\sin^4 x + \cos^4 x}{\sin x \cos x}\]
\[ = \frac{\left( \sin^2 x \right)^2 + \left( \cos^2 x \right)^2}{\sin x \cos x}\]
\[ = \frac{\left( \sin^2 x + \cos^2 x \right)^2 - 2 \sin^2 x \cos^2 x}{\sin x \cos x}\]
\[ = \frac{1^2 - 2 \sin^2 x \cos^2 x}{\sin x \cos x}\]
\[ = \frac{1 - 2 \sin^2 x \cos^2 x}{\sin x \cos x}\]
 = RHS
Hence proved.

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 5 Trigonometric Functions
Exercise 5.1 | Q 10 | Page 18
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×