Question 7 of 18intermediate🔍 AnalyzeShort Answer2 marks

If 3 cot A = 4, check whether 1−tan⁡2A1+tan⁡2A=cos⁡2A−sin⁡2A\frac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A - \sin^2 A or not.

Correct Answer

Yes, the given equation holds true. Both LHS and RHS equal 7/25.

Exercise: EXERCISE 8.1 | Q: 8 | (Page Page 9)
For More Understanding

Explanation

This question tests the application of trigonometric identities and the relationship between different trigonometric ratios. The student needs to find sin A and cos A from the given cot A value, then verify the equality.

Solution Steps

  1. Step 1: Given 3 cot A = 4, therefore cot A = 4/3 and tan A = 3/4.

  2. Step 2: Calculate LHS: tan²A = 9/16. Using identity 1 + tan²A = sec²A, we get 1 + tan²A = 25/16 and 1 - tan²A = 7/16. So LHS = (7/16)/(25/16) = 7/25.

  3. Step 3: Calculate RHS: Since cot A = cos A/sin A = 4/3, let cos A = 4k and sin A = 3k. Using sin²A + cos²A = 1, we get k = 1/5. Thus sin A = 3/5 and cos A = 4/5.

  4. Step 4: RHS = cos²A - sin²A = 16/25 - 9/25 = 7/25.

  5. Step 5: Since LHS = RHS = 7/25, the equation is verified.

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