Ex 8.3, 4 (i) - Chapter 8 Class 10 Introduction to Trignometry

Transcript

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