CBSE · Class 10 · Mathematics · Introduction to TrigonometryProve the following trigonometric identity: $$\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$$\nAlso, explain the fundamental trigonometric identities used in proving such identities.

Proof of the Identity

\nGiven expression: $$\text{LHS} = \frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta}$$

Step 1: Factor out $\sin \theta$ from the numerator and $\cos \theta$ from the denominator. $$\text{LHS} = \frac{\sin \theta (1 - 2 \sin^2 \theta)}{\cos \theta (2 \cos^2 \theta - 1)}$$

Step 2: Express $\sin^2 \theta$ and $\cos^2 \theta$ in terms of a single trigonometric ratio using the fundamental identity $\sin^2 \theta + \cos^2 \theta = 1$.\nWe know that $\sin^2 \theta = 1 - \cos^2 \theta$. Substituting this in the numerator: $$\text{Numerator} = \sin \theta (1 - 2(1 - \cos^2 \theta))$$ $$\text{Numerator} = \sin \theta (1 - 2 + 2 \cos^2 \theta)$$ $$\text{Numerator} = \sin \theta (2 \cos^2 \theta - 1)$$

Step 3: Substitute the simplified numerator back into the fraction. $$\text{LHS} = \frac{\sin \theta (2 \cos^2 \theta - 1)}{\cos \theta (2 \cos^2 \theta - 1)}$$

Step 4: Cancel out the common factor $(2 \cos^2 \theta - 1)$ from both numerator and denominator. $$\text{LHS} = \frac{\sin \theta}{\cos \theta}$$

Step 5: Use the quotient relation $\frac{\sin \theta}{\cos \theta} = \tan \theta$. $$\text{LHS} = \tan \theta = \text{RHS}$$\nHence, proved.

Explanation of Fundamental Trigonometric Identities

\nTrigonometric identities are equations involving trigonometric ratios that are true for all values of the angles involved. The study of introduction to trigonometry relies heavily on three primary Pythagorean identities, reciprocal relations, and quotient relations.

• Pythagorean Identities:

  1. $\sin^2 \theta + \cos^2 \theta = 1$
  2. $1 + \tan^2 \theta = \sec^2 \theta$
  3. $1 + \cot^2 \theta = \ cosec^2 \theta$ These identities are derived directly from the Pythagorean theorem applied to a right-angled triangle where the sides represent perpendicular, base, and hypotenuse.

• Quotient Relations:

  1. $\tan \theta = \frac{\sin \theta}{\cos \theta}$
  2. $\cot \theta = \frac{\cos \theta}{\sin \theta}$ These relations connect the tangent and cotangent ratios directly to sine and cosine, making complex fraction simplifications much easier during proofs.

• Reciprocal Relations:

  1. $\sin \theta = \frac{1}{\csc \theta}$
  2. $\cos \theta = \frac{1}{\sec \theta}$
  3. $\tan \theta = \frac{1}{\cot \theta}$ Reciprocal relations establish the inverse relationship between the primary trigonometric ratios and their corresponding secondary ratios.