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受欢迎的三角函数 >

sec^2(arctan(2))-csc^2(arctan(3))

解答

sec^{2} (arctan (2)) - csc^{2} (arctan (3))

解答

\frac{35}{9}

+1

十进制

3.88888 \dots

求解步骤

sec^{2} (arctan (2)) - csc^{2} (arctan (3))

使用三角恒等式改写:

csc (arctan (3)) = \frac{1 + 3 ^{2}}{3}

利用以下特性:

csc (arctan (x)) = \frac{1 + x ^{2}}{x}

= \frac{1 + 3 ^{2}}{3}

= sec^{2} (arctan (2)) - (\frac{1 + 3 ^{2}}{3})^{2}

化简

= sec^{2} (arctan (2)) - \frac{10}{9}

使用三角恒等式改写:

sec (arctan (2)) = 5

sec (arctan (2))

使用三角恒等式改写:

sec (arctan (2)) = 1 + 2^{2}

利用以下特性:

sec (arctan (x)) = 1 + x^{2}

= 1 + 2^{2}

= 1 + 2^{2}

化简

= 5

= (5)^{2} - \frac{10}{9}

化简

(5)^{2} - \frac{10}{9} : \frac{35}{9}

(5)^{2} - \frac{10}{9}

(5)^{2} = 5

(5)^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= (5^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 5^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 5

= 5 - \frac{10}{9}

将项转换为分式:

5 = \frac{5 \cdot 9}{9}

= \frac{5 \cdot 9}{9} - \frac{10}{9}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{5 \cdot 9 - 10}{9}

5 \cdot 9 - 10 = 35

5 \cdot 9 - 10

数字相乘:

5 \cdot 9 = 45

= 45 - 10

数字相减:

45 - 10 = 35

= 35

= \frac{35}{9}

= \frac{35}{9}

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