2018 · Q66
Consider the following statements :
- 1²θ = 1 - p² + a²/2pq , where p, q are non-zero real numbers, is possible only when p = q.
- 2²θ = 4pq/(p+q)² - 1 , where p, q are non-zero real numbers, is possible only when p = q.
Which of the statements given above is/are correct ?
Elementary MathematicsTrigonometry
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Statement 1: cos^2(theta) = 1 - (p^2 + q^2)/(2pq) = -(p - q)^2/(2pq); for this to lie in [0, 1] we need p = q (any p != q makes it negative or greater than 1): true. Statement 2: tan^2(theta) = 4pq/(p + q)^2 - 1 = -(p - q)^2/(p + q)^2 <= 0, and since tan^2(theta) >= 0, equality to 0 forces p = q: true. Both statements are correct. Hence option C.