JEE Main Maths: Trigonometric Ratios and Identities questions with solutions

Q1. In triangle ABC, the ratio tanA: tanB: tanC = 1: 2: 1. Find the value of tanA*tanC + tanB*tanC + tanA*tanB.

1. 10
2. 8
3. 6
4. 12

Answer: 10

With tanA=t, tanB=2t, tanC=t, the identity gives 4t = 2t³, so t²=2. Then tanA*tanC + tanB*tanC + tanA*tanB = t² + 2t² + 2t² = 5t² = 10.

Q2. Is the following identity correct? sin(45deg + A) * sin(45deg - A) = (1/2) cos 2A.

1. True
2. False
3. Cannot be determined
4. None of these

Answer: True

Applying the product-to-difference formula gives (1/2)[cos2A - cos90deg] = (1/2)cos2A, so the identity holds.

Q3. Evaluate the product sin12deg * sin48deg * sin54deg.

1. 1/8
2. 1/4
3. sqrt(3)/8
4. 1/16

Answer: 1/8

Combining the product gives a telescoping use of cos36 and cos72 values that simplifies exactly to 1/8.

Q4. Evaluate sin²(72deg) - sin²(60deg).

1. (sqrt(5) - 1)/8
2. (1 + sqrt(5))/8
3. (3 + sqrt(5))/16
4. (sqrt(5) - 1)/16

Answer: (sqrt(5) - 1)/8

The expression equals sin(132deg)sin(12deg); evaluating numerically gives 0.1545 = (sqrt5 - 1)/8.