e \( \frac{\sin 3 A+\sin A}{2 \sin 2 A} \equiv \cos A \) f \( \frac{\cos 3 A-\sin 3 A}{1-2 \sin 2 A} \equiv \cos A+\sin A \) g \( \quad \frac{\cos 2 A+9 \cos A+5}{4+\cos A} \equiv 2 \cos A+1 \quad \) h \( \quad \frac{\cos ^{3} A-\sin ^{3} A}{\cos A-\sin A} \equiv \frac{2+\sin 2 A}{2} \) P 3 Use the fact that \( 4 A=2 \times 2 A \) to show that: a \( \frac{\sin 4 A}{\sin A} \equiv 8 \cos ^{3} A-4 \cos A \) b \( \cos 4 A+4 \cos 2 A \equiv 8 \cos ^{4} A-3 \) P 4 Prove the identity \( 8 \sin ^{2} x \cos ^{2} x \equiv 1-\cos 4 x \). P 5 Prove the identity \( (2 \sin A+\cos A)^{2} \equiv \frac{1}{2}(4 \sin 2 A-3 \cos 2 A+5) \). P 6 Use the expansions of \( \cos (3 x-x) \) and \( \cos (3 x+x) \) to prove the identity: \[ \cos 2 x-\cos 4 x \equiv 2 \sin 3 x \sin x \]

Let's tackle **Problem P6**, which involves proving the trigonometric identity: \[ \cos 2x - \cos 4x \equiv 2 \sin 3x \sin x \] To prove this identity, we'll use the **cosine of sum and difference formulas**, as well as a product-to-sum identity. Here's a step-by-step breakdown: ### Step 1: Express \(\cos 2x\) and \(\cos 4x\) Using Sum and Difference Formulas First, recognize that \(\cos 2x\) and \(\cos 4x\) can be expressed in terms of \(\cos(3x - x)\) and \(\cos(3x + x)\): \[ \cos(3x - x) = \cos 2x \] \[ \cos(3x + x) = \cos 4x \] ### Step 2: Apply the Cosine Subtraction Formula Recall the **cosine subtraction formula**: \[ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \] Apply this formula to \(\cos 2x - \cos 4x\): \[ \cos 2x - \cos 4x = -2 \sin\left(\frac{2x + 4x}{2}\right) \sin\left(\frac{2x - 4x}{2}\right) \] Simplify the arguments of the sine functions: \[ = -2 \sin(3x) \sin(-x) \] ### Step 3: Simplify Using Sine's Odd Function Property Remember that \(\sin(-x) = -\sin x\). Apply this property: \[ = -2 \sin(3x) (-\sin x) \] \[ = 2 \sin 3x \sin x \] ### Conclusion We've successfully shown that: \[ \cos 2x - \cos 4x = 2 \sin 3x \sin x \] Therefore, the identity is proven. \[ \boxed{\cos 2x - \cos 4x \equiv 2 \sin 3x \sin x} \]