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Misc 10 - Prove cot-1 ( root (1 + sin x) + root (1 - sin x)) - Miscellaneous

  1. Chapter 2 Class 12 Inverse Trigonometric Functions
  2. Serial order wise
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Misc 10 Prove cot-1 ((√(1 + sin⁡〖x 〗 ) + √(1 − sin⁡x ))/(√(1 +〖 sin〗⁡x ) − √(1 − sin⁡x ))) = 𝑥/2 , x ∈ (0, 𝜋/4)  First, finding √(1+sin⁡𝑥 ) & √(1−sin⁡𝑥 ) cot-1 ((√(1 + sin⁡x ) + √(1 − sin⁡x ))/(√(1 + sin⁡x ) − √(1 −〖 sin〗⁡x ))) = cot-1 (("(sin " 𝑥/2 " + cos " 𝑥/2 )+" cos " 𝑥/2 "− sin " 𝑥/2)/("sin " 𝑥/2 " + cos " 𝑥/2 " − (cos " 𝑥/2 "− sin " 𝑥/2))) = cot-1 (("sin " 𝑥/2 −" sin " 𝑥/2 +" cos " 𝑥/2 + " cos " 𝑥/2)/("cos " 𝑥/2 "− cos " 𝑥/2 +" sin " 𝑥/2 + "sin " 𝑥/2)) = cot-1 ((0 +〖 2cos〗⁡〖 𝑥/2〗)/(0 + 2sin⁡〖 𝑥/2〗 )) = cot-1 ((2 cos x/2 )/〖2 sin〗⁡〖 x/2 〗 ) = cot-1 (cot x/2) = x/2 = R.H.S.

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