Question 33 of 50intermediate🔧 ApplyLong Answer4 marks

Prove that:

cos⁡9x−cos⁡5xsin⁡17x−sin⁡3x=−sin⁡2xcos⁡10x\frac{\cos 9x - \cos 5x}{\sin 17x - \sin 3x} = -\frac{\sin 2x}{\cos 10x}
Correct Answer

We begin with the L.H.S. of the given equation:

L.H.S.=cos⁡9x−cos⁡5xsin⁡17x−sin⁡3x\text{L.H.S.} = \frac{\cos 9x - \cos 5x}{\sin 17x - \sin 3x}

Using the identity cos⁡A−cos⁡B=−2sin⁡A+B2sin⁡A−B2\cos A - \cos B = -2 \sin \frac{A+B}{2} \sin \frac{A-B}{2} for the numerator, where A=9xA = 9x and B=5xB = 5x:

cos⁡9x−cos⁡5x=−2sin⁡(9x+5x2)sin⁡(9x−5x2)=−2sin⁡7xsin⁡2x\cos 9x - \cos 5x = -2 \sin \left( \frac{9x + 5x}{2} \right) \sin \left( \frac{9x - 5x}{2} \right) = -2 \sin 7x \sin 2x

Using the identity sin⁡A−sin⁡B=2cos⁡A+B2sin⁡A−B2\sin A - \sin B = 2 \cos \frac{A+B}{2} \sin \frac{A-B}{2} for the denominator (as shown in Example 16), where A=17xA = 17x and B=3xB = 3x:

sin⁡17x−sin⁡3x=2cos⁡(17x+3x2)sin⁡(17x−3x2)=2cos⁡10xsin⁡7x\sin 17x - \sin 3x = 2 \cos \left( \frac{17x + 3x}{2} \right) \sin \left( \frac{17x - 3x}{2} \right) = 2 \cos 10x \sin 7x

Substituting these simplified forms back into the expression:

L.H.S.=−2sin⁡7xsin⁡2x2cos⁡10xsin⁡7x\text{L.H.S.} = \frac{-2 \sin 7x \sin 2x}{2 \cos 10x \sin 7x}

Cancelling the common terms 22 and sin⁡7x\sin 7x:

L.H.S.=−sin⁡2xcos⁡10x=−sin⁡2xcos⁡10x\text{L.H.S.} = \frac{-\sin 2x}{\cos 10x} = -\frac{\sin 2x}{\cos 10x}

This matches the R.H.S. Hence proved.

Exercise: EXERCISE 3.3 | Q: 16 | (Page 25)
For More Understanding

Explanation

This question appears as Question 16 in the Miscellaneous Exercise. The solution applies the sum-to-product trigonometric identities.

The numerator uses the identity for the difference of cosines, which introduces the necessary negative sign. The denominator uses the identity for the difference of sines, demonstrated in Example 16 of the provided text.

Simplifying the resulting fraction by cancelling common factors leads to the desired result.

Solution Steps

  1. Step 1: Identify L.H.S. = (cos 9x - cos 5x) / (sin 17x - sin 3x).

  2. Step 2: Apply identity for numerator: cos 9x - cos 5x = -2 sin 7x sin 2x.

  3. Step 3: Apply identity for denominator: sin 17x - sin 3x = 2 cos 10x sin 7x.

  4. Step 4: Substitute into L.H.S.: L.H.S. = (-2 sin 7x sin 2x) / (2 cos 10x sin 7x).

  5. Step 5: Cancel common terms to get L.H.S. = - sin 2x / cos 10x = R.H.S.

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