Trigonometric EquationMedium
Question
sec2 θ =
is true if and only if
is true if and only ifOptions
A.x = y ≠ 0
B.x = y, x ≠ 0
C.x = y
D.x ≠ 0, y ≠ 0
Solution
We know that, θ ≥ 1
⇒
⇒ 4xy ≥ (x + y)2
⇒ (x + y)2 - 4xy ≤ 0
⇒ (x + y)2 ≤ 0
⇒ x - y = 0
⇒ x = y
Therefore, x + y = 2x (add x both sides)
But x + y ≠ 0 since it lies in the denominator,
⇒ 2x ≠ 0
⇒ x ≠ 0
Hence, x = y, x ≠ 0 is the answer.
Therefore, (a) and (b) are the answers.
⇒
⇒ 4xy ≥ (x + y)2
⇒ (x + y)2 - 4xy ≤ 0
⇒ (x + y)2 ≤ 0
⇒ x - y = 0
⇒ x = y
Therefore, x + y = 2x (add x both sides)
But x + y ≠ 0 since it lies in the denominator,
⇒ 2x ≠ 0
⇒ x ≠ 0
Hence, x = y, x ≠ 0 is the answer.
Therefore, (a) and (b) are the answers.
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