Question

докажите тождество 5sin²a+tgacosa+5cos²a=5+sina
ctgasina-2cos²a-2sin²a=cosa-2​

Answer 1

$\displaystyle\bf\\1)\\\\5Sin^{2} \alpha +tg\alpha Cos\alpha +5Cos^{2} \alpha =(5Sin^{2} \alpha+ 5Cos^{2} \alpha )+tg\alpha \cdot Cos\alpha =\\\\\\=5\cdot(\underbrace{Sin^{2} \alpha+ Cos^{2} \alpha}_{1} )+\frac{Sin\alpha }{Cos\alpha } \cdot Cos\alpha =5+Sin\alpha\\\\\\5+Sin\alpha =5+Sin\alpha$

Тождество доказано

$\displaystyle\bf\\2)\\\\Ctg\alpha Sin\alpha -2Cos^{2} \alpha -2Sin^{2} \alpha =Ctg\alpha Sin\alpha -2\cdot(\underbrace{Cos^{2} \alpha +Sin^{2} \alpha}_{1}) =\\\\\\=\frac{Cos\alpha }{Sin\alpha } \cdot Sin\alpha -2=Cos\alpha -2\\\\\\Cos\alpha -2=Cos\alpha -2$

Тождество доказано