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### Geometry8.11 Trignometry Identities

 Consider any acute angle AOB. P is a point on the ray OB and PQ is perpendicular to OA. We have Sin = PQ/OP Cos = OQ/OP Tan = PQ/OQ Cosec = 1/Sin = OP/PQ Sec = 1/Cos = OP/OQ Cot = 1/tan = OQ/OP Important identities : sin2 + cos2 = 1 1 + tan2 = sec2 1 + cot2 = cosec2 Example: Show that cot + tan = sec .cosec cot + tan = cos /sin + sin /cos = cos2 +sin2 /sin cos = 1/sin cos = 1/sin . 1/cos = cosec .sec   Directions: Find the mid point of the segment joining two points and draw a graph for the problems.
 Q 1: Prove that tan x + cot x = sec x. cosec xAnswer: Q 2: If sin x + cosec x = 2, prove htat sin2 + cosec 2 = 2.Answer: Q 3: Prove that sec4x = sec2x = tan2x + tan4Answer: Q 4: If cos x + sin x = 21/2 cos x, prove that cos x - sin x = 21/2 sin x.Answer: Q 5: Prove that cos x/(sec x - tan x) = 1 + sin xAnswer: Q 6: Prove that (1+tanx)2 + (1-tanx)2 = 2 sec2x.Answer: Q 7: Prove that (sec x + tan x -1)(sec x - tan x + 1) - 2 tan x = 0Answer: Q 8: If tan x + sin x = m and tan x - sin x = n, show that m2 - n2 = 4mn1/2Answer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!