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Ex 3.3, 8 - Prove cos (pi + x) ... - Chapter 3 Class 11 Trigonometry

Ex 3.3, 8 - Chapter 3 Class 11 Trigonometric Functions - Part 2

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Ex 3.3, 8 Prove that cos⁑〖 (Ο€ + π‘₯) cos⁑〖(βˆ’ π‘₯)γ€— γ€—/(sin⁑〖 (Ο€ βˆ’ π‘₯)γ€— cos⁑〖"(" Ο€/2 " + " π‘₯")" γ€— ) = cot2 π‘₯ Taking L.H.S. π‘π‘œπ‘ β‘γ€–(πœ‹ + π‘₯) γ€–π‘π‘œπ‘  〗⁑〖(βˆ’π‘₯)γ€— γ€—/(𝑠𝑖𝑛⁑(πœ‹ βˆ’ π‘₯) π‘π‘œπ‘ β‘γ€–"(" πœ‹/2 " + " π‘₯")" γ€— ) Putting Ο€ = 180Β° = π‘π‘œπ‘ β‘γ€–(180Β° + π‘₯) γ€–π‘π‘œπ‘  〗⁑〖(βˆ’π‘₯)γ€— γ€—/(𝑠𝑖𝑛⁑(180Β° βˆ’ π‘₯) π‘π‘œπ‘ β‘γ€–"(" 90Β° "+ " π‘₯")" γ€— ) Using cos (180Β° + x) = –cos x cos (βˆ’x) = cos x sin (180Β° – x) = sin x & cos(90Β° + x) = –sin x = (βˆ’π‘π‘œπ‘ β‘γ€–π‘₯ Γ— π‘π‘œπ‘ β‘π‘₯ γ€—)/((sin⁑〖π‘₯) Γ—γ€–(βˆ’sin〗⁑π‘₯) γ€— ) = (βˆ’π‘π‘œπ‘ 2π‘₯)/(βˆ’π‘ π‘–π‘›2π‘₯) = cot2x = R.H.S. Hence proved

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