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Misc 5 - Prove cos-1 4/5 + cos-1 12/13 = cos-1 33/65 - Miscellaneous

Misc. 5 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 2
Misc. 5 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 3

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Misc 5 Prove cos-1 4/5 + cos-1 12/13 = cos-1 33/65 Let a = cos-1 4/5 and b = cos-1 12/13 Finding sin a, cos a Let a = cos-1 4/5 cos a = 4/5 We know that sin a = √(1 −cos^2⁡𝑎 ) =√(1−(4/5)^2 ) " =" √(9/25) "=" 3/5 We use cos (a + b) formula Finding sin b, cos b Let b = cos-1 12/13 cos b = 12/13 We know that sin b = √(1 −𝑐𝑜𝑠2 𝑏) = √(1 −(12/13)^2 ) = √(25/169) = 5/13 Now, cos (a + b) = cos a cos b – sin a sin b cos (a + b) = 4/5 × 12/13 – 3/5 × 5/13 = 48/65 − 3/13 = (48 − 3(5))/65 = (48 − 15)/65 = 𝟑𝟑/𝟔𝟓 Putting cos a = 4/5 , sin a = 3/5 & cos b = 12/13 , sin b = 5/13 Thus, cos (a + b) = 33/65 a + b = cos-1 (33/65) Putting a = cos-1 4/5 & b = cos-1 12/15 cos-1 𝟒/𝟓 + cos-1 𝟏𝟐/𝟏𝟓 = cos-1 𝟑𝟑/𝟔𝟓 Which is what we had to prove Hence proved

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.