Question

(16,17) тригонометрия

Answer 1

16)
$frac{1 + {tg}^{4}a }{ {tg}^{2}a + {ctg}^{2}a } - {tg}^{2} a = frac{ frac{ {cos}^{4} a + {sin}^{4}a }{ {cos}^{4}a } }{ frac{ {sin}^{2} a}{ {cos}^{2}a } + frac{ {cos}^{2}a }{ {sin}^{2} a} } - {tg}^{2}a = frac{frac{ {cos}^{4} a + {sin}^{4}a }{ {cos}^{4}a }}{ frac{ {sin}^{4}a + {cos}^{4} a }{ {sin}^{2}a times {cos}^{2}a } } - {tg}^{2} a = frac{ {sin}^{2} a times {cos}^{2}a }{ {cos}^{4}a } - {tg}^{2}a = frac{ {sin}^{2}a }{ {cos}^{2} a} - {tg}^{2} a = {tg}^{2} a - {tg}^{2} a = 0$
17)
${cos}^{2} (30 + x) + {cos}^{2} (30 - x) - {cos}^{2} x = {(cos30 cosx - sin30sinx)}^{2} + (cos30cosx + sin30sinx)^{2} - {cos}^{2} x = ( frac{ sqrt{3} }{2} cosx - frac{1}{2} sinx)^{2} + {( frac{ sqrt{3} }{2} cosx + frac{1}{2}sinx) }^{2} - {cos}^{2} x = frac{3}{4} {cos}^{2} x - frac{ sqrt{3} }{2} cosxsinx + frac{1}{4} {sin}^{2} x + frac{3}{4} {cos}^{2} x + frac{ sqrt{3} }{2} cosxsinx + frac{1}{4} {sin}^{2} x - {cos}^{2} x = frac{6}{4} {cos}^{2} x + frac{2}{4} {sin}^{2} x - {cos}^{2} x = frac{2}{4} {cos}^{2} x + frac{2}{4} {sin}^{2} x = frac{2}{4} ( {cos}^{2} x + {sin}^{2} x) = frac{2}{4} = frac{1}{2} = 0.5$