Example 10 - Chapter 8 Class 10 Introduction to Trignometry

Transcript

Example 10 Prove that sec A (1 – sin A)(sec A + tan A) = 1. Solving L.H.S sec A (1 – sin A) (sec A + tan A) Writing everything in terms of sin A and cos A = 1/cos⁡〖 𝐴〗 (1 – sin A) (" " 1/cos⁡〖 𝐴〗 +" " sin⁡〖 𝐴〗/cos⁡〖 𝐴〗 ) = ((1 −〖 sin〗⁡〖 𝐴)〗)/(cos⁡ 𝐴) (" " (1 +〖 sin〗⁡〖 𝐴〗)/cos⁡〖 𝐴〗 ) = ((𝟏 −〖 𝒔𝒊𝒏〗⁡〖 𝑨) (𝟏+〖 𝒔𝒊𝒏〗⁡〖 𝑨)〗 〗)/(𝒄𝒐𝒔⁡ 𝑨 × 𝒄𝒐𝒔⁡〖 𝑨〗 ) Since (a – b) (a + b) = a2 – b2 = ((12 −〖 sin2〗⁡〖 𝐴) 〗)/( cos⁡〖2 𝐴〗 ) = ((𝟏 −〖 𝐬𝐢𝐧𝟐〗⁡〖 𝑨) 〗)/( 𝒄𝒐𝒔⁡〖𝟐 𝑨〗 ) = (𝒄𝒐𝒔𝟐 𝑨)/(𝐜𝐨𝐬𝟐⁡ 𝑨) = 1 = R.H.S Thus, L.H.S = R.H.S Hence proved cos2 A + sin2 A = 1 cos2 A = 1 – sin2 A 1 – sin2 A = cos2 A