求证:1/sin²a+1/cos²a-1/tan²a=2+tan²a

问题描述:

求证:1/sin²a+1/cos²a-1/tan²a=2+tan²a

证明:
1/sin²a+1/cos²a-1/tan²a
=(sin²a+cos²a)/sin²a*cos²a-1/tan²a
=1/sin²a*cos²a-1/tan²a
=1/sin²a*cos²a-cos²a/sin²a
=(1/sin²a)[(1/cos²a)-cos²a]
=(1/sin²a*cos²a)[(1-cos²a*cos²a]
=(1/sin²a*cos²a)[(1+cos²a](1-cos²a)]
=(1/cos²a)[1+cos²a]
=(1/cos²a)+1
=sec²a+1
=tan²a+1+1
=tan²a+2.