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

Transcript

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. ((1 +𝑡𝑎𝑛2 𝐴)/(1 + 𝑐𝑜𝑡2 𝐴))=((1 −tan⁡〖 𝐴〗)/(1 −cot⁡ 𝐴))^2=𝑡𝑎𝑛2 𝐴 Solving ((𝟏 + 𝒕𝒂𝒏𝟐 𝑨)/(𝟏 + 𝒄𝒐𝒕𝟐 𝑨)) ((1 + 𝑡𝑎𝑛2 𝐴)/(1 + 𝒄𝒐𝒕𝟐 𝐴)) = ((1 + 𝑡𝑎𝑛2 𝐴))/(((1+ 𝟏/(𝒕𝒂𝒏𝟐 𝑨)) ) = ((1 + 𝑡𝑎𝑛2 𝐴))/(((tan^2⁡𝐴 + 1))/(tan^2⁡𝐴 ))= (𝑡𝑎𝑛2 𝐴 (1 + 𝑡𝑎𝑛2 𝐴))/((𝑡𝑎𝑛2 𝐴 + 1)) = tan2 A = R.H.S Solving ((𝟏− 𝒕𝒂𝒏⁡𝑨)/(𝟏− 𝒄𝒐𝒕⁡𝑨 ))^𝟐 ((1− tan⁡𝐴)/(1− 𝒄𝒐𝒕⁡𝑨 ))^2 = ((1 − tan⁡〖 𝐴〗)/(1 − 𝟏/𝒕𝒂𝒏⁡〖 𝑨〗 ) " " )^2 = (((1 − tan⁡〖 𝐴)〗)/(((tan⁡〖 𝐴 −1〗 ))/tan⁡〖 𝐴〗 ))^2 = (tan⁡〖 𝐴(1 − tan⁡〖 𝐴)〗 〗/( (tan⁡〖 𝐴 −1)〗 ))^2 = (tan⁡〖 𝐴(1 − tan⁡〖 𝐴)〗 〗/(−(1 − tan⁡〖 𝐴)〗 ))^2 = (−tan⁡𝐴 )^2 = tan2 A = RHS Therefore, ((1 + 𝑡𝑎𝑛2 𝐴)/(1 + 𝑐𝑜𝑡2 𝐴))=((1 − tan⁡〖 𝐴〗)/(1 − cot⁡ 𝐴))^2=𝑡𝑎𝑛2 𝐴 H\ence proved