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Maths · Trigonometrical identities and trigonometrical functions

If \) and then

If \( \sin x+\cos x=\sqrt{y+\frac{1}{y}}, x \epsilon[0, \pi] \) and \( \boldsymbol{y}>0, \) then

  • A. \( x=\pi / 4 \)
  • B. \( x=\frac{\pi}{2} \)
  • C. \( x=\frac{\pi}{6} \)
  • D. \( x=3 \pi / 4 \)

Step-by-step solution

The left side sin x + cos x has a maximum value √2 when x = π/4 (since sin x+cos x = √2 sin(x+π/4)). The right side √(y+1/y) has minimum value √2 (by AM-GM, y+1/y ≥ 2, equality at y=1). Thus equality holds only when both sides equal √2, requiring y=1 and sin x+cos x = √2, which occurs at x=π/4 within [0,π].
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