三角函數計算

1) sin(π -x) sin(x -3π/2) tan(x -π/2)
─────────────────────
sin(π/2 +x) tan(x +3π/2) tan(2π-x)
=sin x cos x (-cot x)
─────────────── = - cos x
cos x (-cot x) (- tan x);sin²(π+x) / sin³(3π/2 +x) + tan(3π/2 +x) / cot(π+x)
= sin²x / -cos³x + (-cot x) / cot x
= - sin²x / cos³x - 1
= - (sin²x + cos³x) / cos³x
∴ 原式
= ( - cos x ) [- (sin²x + cos³x) / cos³x]
= tan² x + cos x
2)(2 - cos²x)(1 + 2cot²x)
= (2 - cos²x) (1 + 2cos²x / sin²x)
= (2 - cos²x) (1 + cos²x) / sin²x
= (2 - cos²x) (2 + cos²x - 1) / sin²x
= (2 - cos²x) (2 - sin²x) / sin²x
= (2sin²x + 2cos²x - cos²x) (2 - sin²x) / sin²x
= (2sin²x + cos²x) (2 - sin²x) / sin²x
= (2 - sin² x) (2 + cot² x)

3)右式
= cosx / (1 + sinx) - sinx / (1 + cosx)
= [(cosx + cos²x) - (sinx + sin²x)] / [(1 + sinx) (1 + cosx)]
= (cosx - sinx + cos²x - sin²x) / (1 + sinx + cosx + sinx cosx)
= 2(cosx - sinx) (1 + cosx + sinx) / [2(sin²x + cos²x + sinx + cosx + sinx cosx)]
= 2(cosx - sinx) (1 + cosx + sinx) / (sin²x+2sinxcosx+cos²x + 2(sinx + cosx) + 1)
= 2(cosx - sinx) (1 + cosx + sinx) / (1 + cosx + sinx)²
= 2(cosx - sinx) / (1 + cosx + sinx)
= 左式