Biểu thức \(\tan\left(\dfrac{7\pi}{2}-\alpha\right)+\cot\left(\dfrac{11\pi}{2}-\alpha\right)-\tan\left(\dfrac{15\pi}{2}+\alpha\right)-\cot\left(\dfrac{19\pi}{2}+\alpha\right)\) có giá trị bằng giá trị của biểu thức nào sau đây?
\(\tan\alpha+\cot\alpha\).\(2(\tan\alpha+\cot\alpha)\).\(2(\tan\alpha-\cot\alpha)\).\(\tan\alpha-\cot\alpha\).Hướng dẫn giải:Ta có: \(\dfrac{7\pi}{2}-\alpha=\dfrac{\pi}{2}-\alpha+3\pi\) nên \(\tan\left(\dfrac{7\pi}{2}-\alpha\right)=\tan\left(\dfrac{\pi}{2}-\alpha\right)=\cot\alpha\)
\(\dfrac{11\pi}{2}-\alpha=\dfrac{\pi}{2}-\alpha+5\pi\) nên \(\cot\left(\dfrac{11\pi}{2}-\alpha\right)=\cot\left(\dfrac{\pi}{2}-\alpha\right)=\tan\alpha\)
Tương tự: \(\tan\left(\dfrac{15\pi}{2}+\alpha\right)=\cot\left(-\alpha\right)=-\cot\alpha\)
\(\cot\left(\dfrac{19\pi}{2}+\alpha\right)=\tan\left(-\alpha\right)=-\tan\alpha\)
Suy ra: \(\tan\left(\dfrac{7\pi}{2}-\alpha\right)+\cot\left(\dfrac{11\pi}{2}-\alpha\right)-\tan\left(\dfrac{15\pi}{2}+\alpha\right)-\cot\left(\dfrac{19\pi}{2}+\alpha\right)\)
\(=\cot\alpha+\tan\alpha-\left(-\cot\alpha\right)-\left(-\tan\alpha\right)\) = \(2(\tan\alpha+\cot\alpha)\)