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secondary 4 | A Maths
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How to prove this identity
Date Posted: 3 years ago
Views: 144
cosec² 2x + cot 2x cosec 2x
= 1/sin² 2x + (cos 2x / sin 2x)(1/sin 2x)
= 1/sin² 2x + cos 2x / sin² 2x
= (1 + cos 2x) / sin² 2x
= (1 + 2 cos²x - 1) / (2 sin x cos x)²
[Recall that cos 2x = cos² x - sin² x = 2 cos² x - 1 = 1 - 2 sin² x]
= 2 cos² x / 4 sin² x cos² x
= 1 / 2 sin² x
= ½ (1/sin² x)
= ½ cosec² x (proved)
= 1/sin² 2x + (cos 2x / sin 2x)(1/sin 2x)
= 1/sin² 2x + cos 2x / sin² 2x
= (1 + cos 2x) / sin² 2x
= (1 + 2 cos²x - 1) / (2 sin x cos x)²
[Recall that cos 2x = cos² x - sin² x = 2 cos² x - 1 = 1 - 2 sin² x]
= 2 cos² x / 4 sin² x cos² x
= 1 / 2 sin² x
= ½ (1/sin² x)
= ½ cosec² x (proved)
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