# Ex 5.6, 11

Last updated at March 11, 2017 by Teachoo

Last updated at March 11, 2017 by Teachoo

Transcript

Ex 5.6, 11 If 𝑥= 𝑎 𝑠𝑖𝑛−1𝑡 , y= 𝑎 𝑐𝑜𝑠−1𝑡, show that 𝑑𝑦𝑑𝑥 = − 𝑦𝑥 𝑥= 𝑎 𝑠𝑖𝑛−1𝑡 , y= 𝑎 𝑐𝑜𝑠−1𝑡 We need to show 𝑑𝑦𝑑𝑥 = − 𝑦𝑥 𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑥 × 𝑑𝑡𝑑𝑡 𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑡 × 𝑑𝑡𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑡 𝑑𝑥𝑑𝑡 Calculating 𝒅𝒚𝒅𝒕 𝑦 = 𝑎 𝑐𝑜𝑠−1𝑡 𝑑𝑦𝑑𝑡 = 𝑑 𝑎 𝑐𝑜𝑠−1𝑡𝑑𝑡 𝑑𝑦𝑑𝑡 = 12 𝑎 𝑐𝑜𝑠−1𝑡 . 𝑑 𝑎 𝑐𝑜𝑠−1𝑡𝑑𝑡 𝑑𝑦𝑑𝑡 = 12 𝑎 𝑐𝑜𝑠−1𝑡 . 𝑎 𝑐𝑜𝑠−1𝑡. log𝑎 × 𝑑 𝑐𝑜𝑠−1𝑡𝑑𝑡 𝑑𝑦𝑑𝑡 = 12 𝑎 𝑐𝑜𝑠−1𝑡 . 𝑎 𝑐𝑜𝑠−1𝑡. log𝑎 × −1 1 − 𝑡2 𝑑𝑦𝑑𝑡 = − 𝑎 𝑐𝑜𝑠−1𝑡. log𝑎 2 𝑎 𝑐𝑜𝑠−1𝑡 . 1 − 𝑡2 Calculating 𝒅𝒙𝒅𝒕 𝑥= 𝑎 𝑠𝑖𝑛−1𝑡 𝑑𝑥𝑑𝑡 = 𝑑 𝑎 𝑠𝑖𝑛−1𝑡𝑑𝑡 𝑑𝑥𝑑𝑡 = 12 𝑎 𝑠𝑖𝑛−1𝑡 × 𝑑 𝑎 𝑠𝑖𝑛−1𝑡𝑑𝑡 𝑑𝑥𝑑𝑡 = 12 𝑎 𝑠𝑖𝑛−1𝑡 × 𝑎 𝑠𝑖𝑛−1𝑡. log𝑎 . 𝑑 𝑠𝑖𝑛−1𝑡𝑑𝑡 𝑑𝑥𝑑𝑡 = 12 𝑎 𝑠𝑖𝑛−1𝑡 × 𝑎 𝑠𝑖𝑛−1𝑡. log𝑎 . 1 1 − 𝑡2 𝑑𝑥𝑑𝑡 = 𝑎 𝑠𝑖𝑛−1𝑡2 𝑎 𝑠𝑖𝑛−1𝑡 × log𝑎 1 − 𝑡2 𝑑𝑥𝑑𝑡 = 𝑎 𝑠𝑖𝑛−1𝑡 . log𝑎2 1 − 𝑡2 Therefore, 𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑡 𝑑𝑥𝑑𝑡 𝑑𝑦𝑑𝑥 = − 𝑎 𝑐𝑜𝑠−1𝑡 . log𝑎 2 1 − 𝑡2 𝑎 𝑠𝑖𝑛−1𝑡 . log𝑎2 1 − 𝑡2 𝑑𝑦𝑑𝑥 = − 𝑎 𝑐𝑜𝑠−1𝑡 . − log𝑎 2 1 − 𝑡2 × 2 1 − 𝑡2 𝑎 𝑠𝑖𝑛−1𝑡 . log𝑎 𝑑𝑦𝑑𝑥 = − 𝑎 𝑐𝑜𝑠−1𝑡 𝑎 𝑠𝑖𝑛−1𝑡 × log𝑎 × 2 1 − 𝑡2 log𝑎 × 2 1 − 𝑡2 𝑑𝑦𝑑𝑥 = − 𝑎 𝑐𝑜𝑠−1𝑡 𝑎 𝑠𝑖𝑛−1𝑡 𝒅𝒚𝒅𝒙 = −𝒚𝒙 Hence proved.

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CA Maninder Singh

CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .