求解 (frac{tan25^{circ}}{csc65^{circ}})^{2}+(frac{cot25^{circ}}{sec65^{circ}})^2+2tan18^circtan45^circtan72^circ

$$( \frac { \tan 25 ^ { \circ } } { \csc 65 ^ { \circ } } ) ^ { 2 } + ( \frac { \cot 25 ^ { \circ } } { \sec 65 ^ { \circ } } ) ^ { 2 } + 2 \tan 18 ^ { \circ } \tan 45 ^ { \circ } \tan 72 ^ { \circ }$$

求值

$$\frac{45442351761115914888034412436150224210730415685514322476705801937969715985475404991335325292689}{15147450587038641294478415595498206962310364838504310844441344450000000000000000000000000000000}\approx 3$$

求解步骤

对题中的三角函数求值

$${(\frac{0.4663076581549986}{1.1033779189624917})} ^ {2} + {(\frac{2.1445069205095586}{2.3662015831524985})} ^ {2} + 2 \cdot 0.3249196962329063 \tan(45) 3.0776835371752527$$

将分子和分母同时乘以 10000000000000000 以展开 \frac{0.4663076581549986}{1.1033779189624917}。

$$\left(\frac{4663076581549986}{11033779189624917}\right)^{2}+\left(\frac{2.1445069205095586}{2.3662015831524985}\right)^{2}+0.3249196962329063\times 2\times 3.0776835371752527\tan(45)$$

通过求根和消去 3，将分数 \frac{4663076581549986}{11033779189624917} 降低为最简分数。

$$\left(\frac{1554358860516662}{3677926396541639}\right)^{2}+\left(\frac{2.1445069205095586}{2.3662015831524985}\right)^{2}+0.3249196962329063\times 2\times 3.0776835371752527\tan(45)$$

计算 2 的 \frac{1554358860516662}{3677926396541639} 乘方，得到 \frac{2416031467266655914917579622244}{13527142578377765566699460806321}。

$$\frac{2416031467266655914917579622244}{13527142578377765566699460806321}+\left(\frac{2.1445069205095586}{2.3662015831524985}\right)^{2}+0.3249196962329063\times 2\times 3.0776835371752527\tan(45)$$

将分子和分母同时乘以 10000000000000000 以展开 \frac{2.1445069205095586}{2.3662015831524985}。

$$\frac{2416031467266655914917579622244}{13527142578377765566699460806321}+\left(\frac{21445069205095586}{23662015831524985}\right)^{2}+0.3249196962329063\times 2\times 3.0776835371752527\tan(45)$$

计算 2 的 \frac{21445069205095586}{23662015831524985} 乘方，得到 \frac{459890993211339028795067396683396}{559890993211339027323350679250225}。

$$\frac{2416031467266655914917579622244}{13527142578377765566699460806321}+\frac{459890993211339028795067396683396}{559890993211339027323350679250225}+0.3249196962329063\times 2\times 3.0776835371752527\tan(45)$$

\frac{2416031467266655914917579622244}{13527142578377765566699460806321} 与 \frac{459890993211339028795067396683396}{559890993211339027323350679250225} 相加，得到 \frac{7573725293519320826744653583305434733489071583881254791964551016}{7573725293519320647239207797749103481155182419252155422220672225}。

$$\frac{7573725293519320826744653583305434733489071583881254791964551016}{7573725293519320647239207797749103481155182419252155422220672225}+0.3249196962329063\times 2\times 3.0776835371752527\tan(45)$$

将 0.3249196962329063 与 2 相乘，得到 0.6498393924658126。

$$\frac{7573725293519320826744653583305434733489071583881254791964551016}{7573725293519320647239207797749103481155182419252155422220672225}+0.6498393924658126\times 3.0776835371752527\tan(45)$$

将 0.6498393924658126 与 3.0776835371752527 相乘，得到 1.99999999999999938244315951584402。

$$\frac{7573725293519320826744653583305434733489071583881254791964551016}{7573725293519320647239207797749103481155182419252155422220672225}+1.99999999999999938244315951584402\tan(45)$$

从三角函数值表中获取 \tan(45) 的值。

$$\frac{7573725293519320826744653583305434733489071583881254791964551016}{7573725293519320647239207797749103481155182419252155422220672225}+1.99999999999999938244315951584402\times 1$$

将 1.99999999999999938244315951584402 与 1 相乘，得到 1.99999999999999938244315951584402。

$$\frac{7573725293519320826744653583305434733489071583881254791964551016}{7573725293519320647239207797749103481155182419252155422220672225}+1.99999999999999938244315951584402$$

\frac{7573725293519320826744653583305434733489071583881254791964551016}{7573725293519320647239207797749103481155182419252155422220672225} 与 1.99999999999999938244315951584402 相加，得到 \frac{45442351761115914888034412436150224210730415685514322476705801937969715985475404991335325292689}{15147450587038641294478415595498206962310364838504310844441344450000000000000000000000000000000}。

$$\frac{45442351761115914888034412436150224210730415685514322476705801937969715985475404991335325292689}{15147450587038641294478415595498206962310364838504310844441344450000000000000000000000000000000}$$