ysin^3α+xcos^3α=a,ysinα-xcosα=0,求证1/x²+1/y²=1/a&su

ysinα-xcosα=0
x=tanαy
x²=tan²αy²
1/x²=cot²α/y²
1/x²+1/y²
=(1+cot²α)/y²
=1/(sin²αy²) ①
又
ysin³α+xcos³α=a
ysin³α+tanαcos³α*y=a
ysin³α+ysinαcos²α=a
ysinα(sin²α+cos²α)=a
ysinα=a ②
所以
②代入①,得
1/x²+1/y²=1/a²
得证.