Question

求证:(2-cos²θ)(1+2cot²θ)=(2+cot²θ)(2-sin²θ).

Answer 1

证明:左边=(1+sin²θ)(1+2cot²θ)

=1+2cot²θ+sin²θ+2sin²θ·cot²θ

=2+cos²θ+2cot²θ;

右边=4-2sin²θ+2cot²θ-cos²θ

=4-2(1-cos²θ)+2cot²θ-cos²θ

=2+cos²θ+2cot²θ.

∵左边=右边,∴原式成立.