Question

Упростить выражение:
$(1+tg^{2} )cos^{4} \alpha + sin^{4 } \alpha$

Answer 1

$(1+tg^2x)\cdot cos^4x+sin^4x=\frac{1}{cos^2x}\cdot cos^4x+sin^4x=\\\\=cos^2x+sin^4x=cos^2x+(1-cos^2x)^2=\\\\=cos^2x+1-2cos^2x+cos^4x=cos^4x-cos^2x+1\\\\\\\star \; \; 1+tg^2x=1+ \frac{sin^2x}{cos^2x}= \frac{cos^2x+sin^2x}{cos^2x}= \frac{1}{cos^2x}\; \; \star$