Question 18 of 50beginner🔧 ApplyShort Answer2 marks

Prove that: sin² π/6 + cos² π/3 - tan² π/4 = -1/2

Correct Answer

L.H.S. = sin² π/6 + cos² π/3 – tan² π/4

= (1/2)² + (1/2)² – (1)²

= 1/4 + 1/4 – 1

= 1/2 – 1

= –1/2 = R.H.S.

Hence proved.

Exercise: EXERCISE 3.3 | Q: 1 | (Page 25)
For More Understanding

Explanation

This question tests knowledge of standard trigonometric values at specific angles. From Example 10 in the textbook, we see that sin π/6 = 1/2 and sec π/3 = 2 (which means cos π/3 = 1/2). Also, cot π/4 = 1, which means tan π/4 = 1. Students should substitute these values and simplify step by step to reach the right-hand side.

Solution Steps

  1. Step 1: Substitute sin π/6 = 1/2, so sin² π/6 = (1/2)² = 1/4

  2. Step 2: Substitute cos π/3 = 1/2, so cos² π/3 = (1/2)² = 1/4

  3. Step 3: Substitute tan π/4 = 1, so tan² π/4 = 1

  4. Step 4: Calculate L.H.S. = 1/4 + 1/4 – 1 = 1/2 – 1 = –1/2 = R.H.S.

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