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If secA=x+1/4x,then prove that secA+tanA=2x or 1/2x |
| Answer» {tex}\\begin{array}{l}If\\;secA=x+1/4x,then\\;prove\\;that\\;secA+\\tan A=2x\\;or\\;1/2x\\\\sec\\varnothing=x+\\frac1{4x}\\\\\\tan^2\\varnothing=sec^2\\varnothing-1=\\left(x+\\frac1{4x}\\right)^2-1=\\left(x+\\frac1{4x}\\right)^2-4x\\frac1{4x}=\\left(x-\\frac1{4x}\\right)^2\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\left[\\left(a+b\\right)^2-4ab=\\left(a-b\\right)^2\\right]\\\\\\tan\\varnothing=\\pm\\left(x-\\frac1{4x}\\right)\\\\when\\;\\;\\;\\;\\;\\tan\\varnothing=x-\\frac1{4x}\\;,\\;sec\\varnothing+\\tan\\varnothing=\\left(x+\\frac1{4x}\\right)+\\left(x-\\frac1{4x}\\right)=2x\\\\when\\;\\tan\\varnothing=-\\left(x-\\frac1{4x}\\right)\\\\so\\;sec\\varnothing+\\tan\\varnothing=\\left(x+\\frac1{4x}\\right)\\left(\\frac1{4x}-x\\right)=\\frac1{4x}+\\frac1{4x}=\\frac1{2x}\\\\hence,sec\\varnothing+\\tan\\varnothing\\;is\\;either\\;2x\\;or\\;\\frac1{2x}.\\end{array}{/tex} | |