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| 1. |
T^n = sin^n + cos^n. to prove :- T^3 - T^5 / T^1 = T^5 - T^7/T^3 |
| Answer» We know,{tex}{T_n} = {\\sin ^n}\\theta + {\\cos ^n}\\theta {/tex}{tex}\\therefore {T_3} = {\\sin ^3}\\theta + {\\cos ^3}\\theta {/tex}{tex}{T_5} = {\\sin ^5}\\theta + {\\cos ^5}\\theta {/tex}{tex}{T_1} = \\sin \\theta + \\cos \\theta {/tex}{tex}{T_7} = {\\sin ^7}\\theta + {\\cos ^7}\\theta {/tex}+ cot{tex}\\theta{/tex}\xa0= m and cosec{tex}\\theta{/tex}\xa0- cot{tex}\\theta{/tex}\xa0= n,LHS\xa0{tex} = \\frac{{{T_3} - {T_5}}}{{{T_1}}}{/tex}{tex} = \\frac{{{{\\sin }^3}\\theta + {{\\cos }^3}\\theta - \\left( {{{\\sin }^5}\\theta + {{\\cos }^5}\\theta } \\right)}}{{\\sin \\theta + \\cos \\theta }}{/tex}{tex} = \\frac{{{{\\sin }^3}\\theta + {{\\cos }^3}\\theta - {{\\sin }^5}\\theta - {{\\cos }^5}\\theta }}{{\\sin \\theta + \\cos \\theta }}{/tex}{tex} = \\frac{{{{\\sin }^3}\\theta - {{\\sin }^5}\\theta + {{\\cos }^3}\\theta - {{\\cos }^5}\\theta }}{{\\sin \\theta + \\cos \\theta }}{/tex}{tex} = \\frac{{{{\\sin }^3}\\theta (1 - {{\\sin }^2}\\theta ) + {{\\cos }^3}\\theta (1 - {{\\cos }^2}\\theta )}}{{\\sin \\theta + \\cos \\theta }}{/tex}{tex} = \\frac{{{{\\sin }^3}\\theta {{\\cos }^2}\\theta + {{\\cos }^3}\\theta {{\\sin }^2}\\theta }}{{\\sin \\theta + \\cos \\theta }}{/tex}\xa0{tex}\\left[ \\begin{gathered} \\because 1 - {\\sin ^2}\\theta = {\\cos ^2}\\theta \\hfill \\\\ 1 - {\\cos ^2}\\theta = {\\sin ^2}\\theta \\hfill \\\\ \\end{gathered} \\right]{/tex}{tex} = \\frac{{{{\\sin }^2}\\theta {{\\cos }^2}\\theta (\\sin \\theta + \\cos \\theta )}}{{(\\sin \\theta + \\cos \\theta )}}{/tex}{tex} = {\\sin ^2}\\theta {\\cos ^2}\\theta {/tex}\xa0RHS\xa0{tex} = \\frac{{{T_5} - {T_7}}}{{{T_3}}}{/tex}{tex} = \\frac{{{{\\sin }^5}\\theta + {{\\cos }^5}\\theta - ({{\\sin }^7}\\theta + {{\\cos }^7}\\theta )}}{{\\left( {{{\\sin }^3}\\theta + {{\\cos }^3}\\theta } \\right)}}{/tex}{tex} = \\frac{{{{\\sin }^5}\\theta + {{\\cos }^5}\\theta - {{\\sin }^7}\\theta - {{\\cos }^7}\\theta }}{{({{\\sin }^3}\\theta + {{\\cos }^3}\\theta )}}{/tex}{tex} = \\frac{{{{\\sin }^5}\\theta - {{\\sin }^7}\\theta + {{\\cos }^5}\\theta - {{\\cos }^7}\\theta }}{{{{\\sin }^3}\\theta + {{\\cos }^3}\\theta }}{/tex}{tex} = \\frac{{{{\\sin }^5}\\theta \\left( {1 - {{\\sin }^2}\\theta } \\right) + {{\\cos }^5}\\theta \\left( {1 - {{\\cos }^2}\\theta } \\right)}}{{{{\\sin }^3}\\theta + {{\\cos }^3}\\theta }}{/tex}{tex} = \\frac{{{{\\sin }^5}\\theta {{\\cos }^2}\\theta + {{\\cos }^5}\\theta {{\\sin }^2}\\theta }}{{{{\\sin }^3}\\theta + {{\\cos }^3}\\theta }}{/tex}{tex} = \\frac{{{{\\sin }^2}\\theta {{\\cos }^2}\\theta \\left( {{{\\sin }^3}\\theta + {{\\cos }^3}\\theta } \\right)}}{{\\left( {{{\\sin }^3}\\theta + {{\\cos }^3}\\theta } \\right)}}{/tex}{tex} = {\\sin ^2}\\theta {\\cos ^2}\\theta {/tex}LHS = RHSHence proved. | |