Question

không sử dụng máy tính, chứng minh rằng:

tan19⁰tan33⁰ + tan33⁰ tan38⁰ + tan38⁰tan19⁰ = 1

Answer 1

\(tan19.tan33+tan33.tan38+tan38.tan19\)

\(=tan33\left(tan19+tan38\right)+tan38.tan19\)

\(=\frac{sin33}{cos33}\left(\frac{sin19}{cos19}+\frac{sin38}{cos38}\right)+\frac{sin38.sin19}{cos38.cos19}\)

\(=\frac{sin33}{cos33}\left(\frac{sin19.cos38+sin38.cos19}{cos19.cos38}\right)+\frac{sin38.sin19}{cos38.cos19}\)

\(=\frac{sin33}{cos33}.\frac{sin57}{cos19.cos38}+\frac{sin38.sin19}{cos38.cos19}=\frac{sin33}{cos33}.\frac{cos33}{cos19.cos38}+\frac{\frac{1}{2}\left(cos19-cos57\right)}{cos38.cos19}\)

\(=\frac{2sin33-cos19-cos57}{2cos38.cos19}=\frac{2sin33-cos19-sin33}{2cos38.cos19}=\frac{sin33-cos19}{2cos38.cos19}\)

\(=\frac{cos\left(90-33\right)-cos19}{cos57-cos19}=\frac{cos57-cos19}{cos57-cos19}=1\)