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Show tan 20 tan40 tan60 tan80=3 |
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Answer» tan 20° tan 40° tan 60° tan 80°= (tan 20° tan 40° tan 80°)\xa0{tex}\\sqrt {3}{/tex}\xa0{tex}\\left[\\because \\tan 60^{\\circ}=\\sqrt{3}\\right]{/tex}{tex}=\\left(\\frac{\\sin 20^{\\circ} \\sin 40^{\\circ} \\sin 80^{\\circ}}{\\cos 20^{\\circ} \\cos 40^{\\circ} \\cos 80^{\\circ}}\\right) \\sqrt{3}{/tex}{tex}=\\frac{\\left(2 \\sin 20^{\\circ} \\sin 40^{\\circ}\\right) \\sin 80^{\\circ} \\times \\sqrt{3}}{\\left(2 \\cos 20^{\\circ} \\cos 40^{\\circ}\\right) \\cos 80^{\\circ}}{/tex}Applying{tex}\\Rightarrow{/tex}\xa02 sin A sin B = cos (A - B) - cos (A + B)2 cos A cos B - cos (A + B) + cos (A - B){tex}\\frac{\\left(\\cos \\left(40^{\\circ}-20^{\\circ}\\right)-\\cos \\left(40^{\\circ}+20^{\\circ}\\right)\\right) \\sin 80^{\\circ} \\times \\sqrt{3}}{\\left(\\cos \\left(20^{\\circ}+40^{\\circ}\\right)+\\cos \\left(40^{\\circ}-20^{\\circ}\\right)\\right) \\cos 80^{\\circ}}{/tex}{tex}=\\frac{\\left(\\cos 20^{\\circ}-\\cos 60^{\\circ}\\right) \\sin 80^{\\circ} \\times \\sqrt{3}}{\\left(\\cos 60^{\\circ}+\\cos 20^{\\circ}\\right) \\cos 80^{\\circ}}{/tex}{tex}=\\frac{\\left(\\cos 20^{\\circ}-\\frac{1}{2}\\right) \\sin 80^{\\circ} \\times \\sqrt{3}}{\\left(\\frac{1}{2}+\\cos 20^{\\circ}\\right) \\cos 80^{\\circ}}{/tex}{tex}=\\frac{\\left(2 \\sin 80^{\\circ} \\cos 20^{\\circ}-\\sin 80^{\\circ}\\right) \\sqrt{3}}{\\cos 80^{\\circ}+2 \\cos 20^{\\circ} \\cos 80^{\\circ}}{/tex}{tex}\\Rightarrow{/tex}\xa02 sin A cos B - sin (A + B) + sin (A - B){tex}=\\frac{\\left(\\sin \\left(80^{\\circ}+20^{\\circ}\\right)+\\sin \\left(80^{\\circ}-20^{\\circ} \\right)-\\sin \\theta 0^{\\circ}\\right) \\sqrt{3}}{\\cos 80^{\\circ}+\\left(\\cos \\left(20^{\\circ}+80^{\\circ}\\right)+\\cos \\left(80^{\\circ}-20^{\\circ}\\right)\\right)}{/tex}{tex}=\\frac{\\left(\\sin 100^{\\circ}+\\sin 60^{\\circ}-\\sin 80^{\\circ}\\right) \\sqrt{3}}{\\cos 80^{\\circ}+\\cos 100^{\\circ}+\\cos 60^{\\circ}}{/tex}{tex}=\\frac{\\left(\\sin \\left(180^{\\circ}-80^{\\circ}\\right)+\\frac{\\sqrt{3}}{2}-\\sin 80^{\\circ}\\right) \\sqrt{3}}{\\cos 80^{\\circ}+\\cos \\left(180^{\\circ}-80^{\\circ}\\right)+\\cos 60^{\\circ}}{/tex}{tex}=\\frac{\\left(\\sin 80^{\\circ}+\\frac{\\sqrt{3}}{2}-\\sin 80^{\\circ}\\right) \\sqrt{3}}{\\cos 80^{\\circ}-\\cos 80^{\\circ}+\\cos 60^{\\circ}}{/tex}{tex}=\\frac{\\frac{3}{2}}{\\frac{1}{2}}{/tex} = 3 = RHS Sia rahende mut kar maths is par , copy pe hi thek he |
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