Chứng minh a) 1 + tan ^2alpha = 1/cos^2alpha . b) 1 + cot ^2alpha = 1/sin^2alpha
Chứng minh
a) \(1 + {\tan ^2}\alpha = \frac{1}{{{\rm{co}}{{\rm{s}}^2}\alpha }}\).
b) \(1 + {\cot ^2}\alpha = \frac{1}{{{\rm{si}}{{\rm{n}}^2}\alpha }}\).
Lời giải
a) Ta có \(1 + {\tan ^2}\alpha = 1 + \frac{{{{\sin }^2}\alpha }}{{{\rm{co}}{{\rm{s}}^2}\alpha }} = \frac{{co{{\rm{s}}^2}\alpha + {{\sin }^2}\alpha }}{{{\rm{co}}{{\rm{s}}^2}\alpha }} = \frac{1}{{{\rm{co}}{{\rm{s}}^2}\alpha }}\).
Vậy \(1 + {\tan ^2}\alpha = \frac{1}{{{\rm{co}}{{\rm{s}}^2}\alpha }}\).
b) Ta có \(1 + {\cot ^2}\alpha = 1 + \frac{{{\rm{co}}{{\rm{s}}^2}\alpha }}{{{\rm{si}}{{\rm{n}}^2}\alpha }} = \frac{{{\rm{si}}{{\rm{n}}^2}\alpha + {\rm{co}}{{\rm{s}}^2}\alpha }}{{{\rm{si}}{{\rm{n}}^2}\alpha }} = \frac{1}{{{\rm{si}}{{\rm{n}}^2}\alpha }}\).
Vậy \(1 + {\cot ^2}\alpha = \frac{1}{{{\rm{si}}{{\rm{n}}^2}\alpha }}\).