Rút gọn biểu thức \(A = \sin \left( {\pi  + x} \right) + \cos \left( {\frac{\pi }{2} - x} \right) + \cot \left( {2\pi  - x} \right) + \tan \left( {\frac{{3\pi }}{2} - x} \right)\). Chọn đáp án đúng?

\(\sin \left( {x + \frac{\pi }{4}} \right) = 0\)\( \Leftrightarrow x + \frac{\pi }{4} = k\pi \)\( \Leftrightarrow x =  - \frac{\pi }{4} + k\pi ,k \in \mathbb{Z}\).

Vì \(x \in \left[ {0;2025\pi } \right]\) nên \(0 \le  - \frac{\pi }{4} + k\pi  \le 2025\pi \)\( \Leftrightarrow \frac{1}{4} \le k \le \frac{{8101}}{4}\) mà \(k \in \mathbb{Z}\) nên \(k \in \left\{ {1;2;..;2025} \right\}\).

Khi đó \(S = \frac{{3\pi }}{4} + \frac{{7\pi }}{4} + \frac{{11\pi }}{4} + ... + \frac{{8099\pi }}{4}\)\( = \frac{\pi }{4}\left( {3 + 7 + 11 + ... + 8099} \right)\)\( = \frac{\pi }{4}.\frac{{\left( {3 + 8099} \right).2025}}{2} = \frac{{4051.2025\pi }}{4}\).

Khi đó \(\frac{{4S}}{{2025\pi }} = \frac{4}{{2025\pi }}.\frac{{4051.2025\pi }}{4} = 4051\).

Trả lời: 4051.

\(A = {\left( {\cos \alpha  + \cos \beta } \right)^2} + {\left( {\sin \alpha  + \sin \beta } \right)^2}\)\( = {\cos ^2}\alpha  + 2\cos \alpha \cos \beta  + {\cos ^2}\beta  + {\sin ^2}\alpha  + 2\sin \alpha \sin \beta  + {\sin ^2}\beta \)

\( = 2 + 2\left( {\cos \alpha \cos \beta  + \sin \alpha \sin \beta } \right)\)\( = 2 + 2\cos \left( {\alpha  - \beta } \right)\)\( = 2 + 2\cos \frac{\pi }{3} = 3\).

Trả lời: 3.