Consider the following statements : 1. \(co{s^2}\theta = 1 - \frac{{{p^2} + {q^2}}}{{2pq}},\) where p, q are non-zero real numbers, is possible only when p = q. 2. \({\tan ^2}\theta = \frac{{4pq}}{{{{\left( {p + q} \right)}^2}}} - 1\), where p, q are non-zero real numbers, is possible only when p = q. Which of the statements given above is/are correct?
Consider the following statements : 1. \(co{s^2}\theta = 1 - \frac{{{p^2} + {q^2}}}{{2pq}},\) where p, q are non-zero real numbers, is possible only when p = q. 2. \({\tan ^2}\theta = \frac{{4pq}}{{{{\left( {p + q} \right)}^2}}} - 1\), where p, q are non-zero real numbers, is possible only when p = q. Which of the statements given above is/are correct? Correct Answer Both 1 and 2
Considering statement 1,
The value of cos2 θ lies in the range
The value of 1 – cos2 θ also lies in the range
⇒ 0 ≤ (1 – cos2 θ) ≤ 1
⇒ 0 ≤ (p2 + q2)/2pq ≤ 1
⇒ 0 ≤ (p2 + q2)/2pq
⇒ 0 ≤ (p2 + q2), is always true as square terms are always greater than zero
Similarly,
⇒ (p2 + q2)/2pq ≤ 1
⇒ (p2 + q2) ≤ 2pq
⇒ (p2 + q2) – 2pq ≤ 0
⇒ (p – q)2 ≤ 0
∵ A square term can never be less than zero,
⇒ (p – q)2 = 0
⇒ p = q
⇒ Statement 1 is correct
Considering statement 2,
Given, tan2 θ = 4pq/(p + q)2 – 1
⇒ sec2 θ = 4pq/(p + q)2
⇒ cos2 θ = (p + q)2/4pq
The value of cos2 θ lies in the range
⇒ 0 ≤ (p + q)2/4pq ≤ 1
⇒ 0 ≤ (p + q)2/4pq
⇒ 0 ≤ (p + q)2, is always true as square terms are always greater than zero
Similarly,
⇒ (p + q)2/4pq ≤ 1
⇒ p2 + q2 + 2pq ≤ 4pq
⇒ p2 + q2 – 2pq ≤ 0
⇒ (p – q)2 ≤ 0
∵ A square term can never be less than zero,
⇒ (p – q)2 = 0
⇒ p = q
⇒ Statement 2 is correct
∴ Both statements 1 and 2 are correct
