(a) (i) Prove that : $\frac{\sin \theta - 2\sin^3 \theta}{2\cos^3 \theta - \cos \theta} = \tan \theta$ (ii) Prove that : $\frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{2\sin^2 A - 1}$ OR (b) If $x \sin^3 \theta + y \cos^3 \theta = \sin \theta \cos \theta$ and $x \sin \theta = y \cos \theta$, prove that $x^2 + y^2 = 1$.

Step 1: Simplify the first expression

To prove the first identity, we start by factoring out common terms from the numerator and the denominator. In the numerator, we factor out $\sin \theta$, and in the denominator, we factor out $\cos \theta$.

Step 2: Apply trigonometric identities

We use the identity $\cos 2\theta = 1 - 2\sin^2 \theta$ and $\cos 2\theta = 2\cos^2 \theta - 1$. Substituting these into the expression, we see that the terms in the parentheses are identical.

Step 3: Simplify to tan θ

Since the terms $(1 - 2\sin^2 \theta)$ and $(2\cos^2 \theta - 1)$ are both equal to $\cos 2\theta$, they cancel out (assuming $\cos 2\theta \neq 0$). This leaves us with $\frac{\sin \theta}{\cos \theta}$, which is equal to $\tan \theta$.

Step 4: Combine fractions for the second expression

For the second identity, we combine the two fractions by finding a common denominator, which is $(\sin A - \cos A)(\sin A + \cos A)$. We then cross-multiply the numerators.

Step 5: Expand and simplify the numerator

We expand the squared terms in the numerator. The terms $2\sin A \cos A$ and $-2\sin A \cos A$ cancel each other out. In the denominator, we use the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$.

Step 6: Apply Pythagorean identity and simplify

Using the Pythagorean identity $\sin^2 A + \cos^2 A = 1$ in the numerator, and substituting $\cos^2 A = 1 - \sin^2 A$ in the denominator, we simplify the expression to the desired right-hand side.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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