Question 20 of 50beginner🔧 ApplyShort Answer2 marks

Prove that: cot² π/6 + cosec 5π/6 + 3 tan² π/6 = 6

Correct Answer

L.H.S. = cot² π/6 + cosec 5π/6 + 3 tan² π/6

Step 1: Find cot² π/6 From Example 12, tan π/6 = 1/3\sqrt{3} Therefore, cot π/6 = 3\sqrt{3} So, cot² π/6 = (3\sqrt{3})² = 3

Step 2: Find cosec 5π/6 From Example 10, sin(5π/6) = sin(π - π/6) = sin π/6 = 1/2 Therefore, cosec 5π/6 = 1/sin(5π/6) = 1/(1/2) = 2

Step 3: Find 3 tan² π/6 tan π/6 = 1/3\sqrt{3} So, tan² π/6 = (1/3\sqrt{3})² = 1/3 Therefore, 3 tan² π/6 = 3 × (1/3) = 1

Step 4: Add all values L.H.S. = 3 + 2 + 1 = 6 = R.H.S.

Hence proved.

Exercise: EXERCISE 3.3 | Q: 3 | (Page 25)
For More Understanding

Explanation

This question from Exercise 3.3 tests students' knowledge of trigonometric values at standard angles. The solution uses the fact that tan π/6 = 1/3\sqrt{3} (shown in Example 12), and the property that sin(π - θ) = sin θ to evaluate sin(5π/6). Students must recall that cot θ = 1/tan θ and cosec θ = 1/sin θ. The key insight is recognizing that 5π/6 = π - π/6, allowing direct application of the sine property demonstrated in Example 10.

Solution Steps

  1. Step 1: Evaluate cot² π/6 = (3\sqrt{3})² = 3

  2. Step 2: Evaluate cosec 5π/6 = 1/sin(π - π/6) = 1/(1/2) = 2

  3. Step 3: Evaluate 3 tan² π/6 = 3 × (1/3) = 1

  4. Step 4: Sum all terms: 3 + 2 + 1 = 6

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