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Example 16 - Prove cos 7x + cos 5x / sin 7x - sin 5x = cot x

Example 16 - Chapter 3 Class 11 Trigonometric Functions - Part 2


Transcript

Example 16 Prove that π‘π‘œπ‘ β‘γ€–7π‘₯ + π‘π‘œπ‘ β‘5π‘₯ γ€—/𝑠𝑖𝑛⁑〖7π‘₯ βˆ’ 𝑠𝑖𝑛⁑5π‘₯ γ€— = cot x Taking L.H.S. We solve cos 7x + cos 5x & sin 7x – sin 5x separately cos x + cos y = 2 cos (π‘₯ + 𝑦)/2 cos (π‘₯ βˆ’ 𝑦)/2 Putting x = 7x & y = 5x cos 7x + cos 5x = 2 cos ((7π‘₯ + 5π‘₯)/2) cos ((7π‘₯ βˆ’ 5π‘₯)/2) = 2 cos (12π‘₯/2) cos (2π‘₯/2) = 2 cos 6x cos x sin x – sin y = 2 cos (π‘₯ + 𝑦)/2 sin (π‘₯ βˆ’ 𝑦)/2 Putting x = 7x & y = 5x sin 7x – sin 5x = 2 cos ((7π‘₯ + 5π‘₯)/2) sin((7π‘₯ βˆ’ 5π‘₯)/2) = 2 cos (12π‘₯/2) sin (2π‘₯/2) = 2 cos 6x sin x Now γ€–π‘π‘œπ‘  〗⁑〖7π‘₯ + π‘π‘œπ‘ β‘5π‘₯ γ€—/𝑠𝑖𝑛⁑〖7π‘₯ βˆ’ 𝑠𝑖𝑛⁑5π‘₯ γ€— = (2 γ€– π‘π‘œπ‘  〗⁑〖6x π‘π‘œπ‘ β‘π‘₯ γ€—)/(2 π‘π‘œπ‘ β‘γ€– 6π‘₯ si𝑛⁑π‘₯ γ€— ) = π‘π‘œπ‘ β‘γ€– π‘₯γ€—/𝑠𝑖𝑛⁑〖 π‘₯γ€— = cot x = R.H.S. Hence L.H.S. = R.H.S. 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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.