Question

Помогите с тригонометрией пожалуйста

Answer 1

$1)\; \; tga+\dfrac{cosa}{1+sina}=\dfrac{sina}{cosa}+\dfrac{cosa}{1+sina}=\dfrac{sina+sin^2a+cos^2a}{cosa\cdot (1+sina)}=\dfrac{sina+1}{cosa\cdot (1+sina)}=\\\\=\dfrac{1}{cosa}\\\\\\2)\; \; sinx-tgx\cdot ctgx=0\\\\sinx-1=0\\\\sinx=1\\\\x=\pi n\; ,\; n\in Z$

$3)\; \; \dfrac{ctga}{tga+ctga}=\dfrac{\frac{cosa}{sina}}{\frac{sina}{cosa}+\frac{cosa}{sina}}=\dfrac{cosa}{sina\cdot \frac{sin^2a+cos^2a}{cosa\cdot sina}}=\dfrac{cos^2a}{sin^2a+cos^2a}=\\\\\\=\dfrac{cos^2a}{1}=cos^2a$

$4)\; \; 5sin^2x+2cosx=5-5cos^2x\\\\5sin^2x=2cosx=5(1-cos^2x)\\\\5sin^2x+2cosx=5sin^2x\\\\2cosx=0\\\\cosx=0\\\\x=\frac{\pi}{2}+\pi n\; ,\; n\in Z$