Упростите выражение и докажите тождество. Created by simonluise. algebra-ru

Упростите выражение и докажите тождество...

simonluise @simonluise

September 2021 1 41 Report

Упростите выражение и докажите тождество

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NNNLLL54

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$1)\; \; sinx\cdot cos6x-sin3x\cdot cos4x=\\\\=\frac{1}{2}\cdot (sin7x+sin(-5x))-\frac{1}{2}\cdot (sin7x+sin(-x))=\\\\=\frac{1}{2}\cdot (-sin5x+sinx)=\frac{1}{2}\cdot 2\cdot sin\frac{x-5x}{2}\cdot cos\frac{x+5x}{2}=-sin2x\cdot cos3x$

$2)\; \; sina\cdot sin(\beta -a)+sin^2(\frac{\beta }{2}-a)=\\\\=\frac{1}{2}\cdot \Big (cos(a-(\beta -a))-cos(a+\beta -a)\Big )+\frac{1-cos(2\cdot \frac{\beta }{2}-2a))}{2}=\\\\=\frac{1}{2}\cdot (cos(2a-\beta )-cos\beta )+\frac{1}{2}-\frac{1}{2}\cdot cos(\beta -2a)=\\\\=\frac{1}{2}\cdot cos(2a-\beta )-\frac{1}{2}cos\beta +\frac{1}{2}-\frac{1}{2}\cdot cos(2a-\beta )=\\\\=\frac{1}{2}\cdot (1-cos\beta )=\frac{1-cos\beta }{2}=sin^2\frac{\beta }{2}\\\\\\\star \; \; sinx\cdot cosy=\frac{1}{2}\cdot (sin(x+y)+sin(x-y))$

$sinx\cdot siny=\frac{1}{2}\cdot (cos(x-y)-cos(x+y))\\\\sin^2x=\frac{1-cos2x}{2}\; \; \star$

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