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Ex 8.3, 4 Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) (cosec θ – cot θ)2 = (1 − 𝑐𝑜𝑠" " θ)/(1 + cos⁡θ ) Solving L.H.S (cosec θ – cot θ)2 We need to make it in terms of cos θ & sin θ = (1/sin⁡𝜃 − cos⁡𝜃/sin⁡𝜃 )^2 = ((1 − cos⁡𝜃)/sin⁡𝜃 )^2 = ((1 − cos⁡〖𝜃)2〗)/(𝒔𝒊𝒏𝟐 𝜽) = ((1 − 〖cos 𝜃〗⁡〖)2〗)/(1 − 𝑐𝑜𝑠2𝜃) = ((1 − 〖cos 𝜃〗⁡〖)2〗)/(1 − 𝑐𝑜𝑠2𝜃) = ((1 −〖 cos 𝜃〗⁡〖)2〗)/(𝟏𝟐 − 𝒄𝒐𝒔𝟐𝜽) = (1− cos 𝜃)^2/((𝟏 + 𝐜𝐨𝐬⁡𝜽)(𝟏 − 𝐜𝐨𝐬⁡𝜽)) = (𝟏 −〖 𝐜𝐨𝐬〗⁡𝜽)/(𝟏 +〖 𝐜𝐨𝐬〗⁡𝜽 ) = RHS Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo