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\(A=-2cosx+2cosx+tan^2x-\frac{1}{cos^2x}\)
\(=tan^2x-\left(1+tan^2x\right)=-1\)
2sin(π2+x)+sin(3π−x)+sin(3π2+x)+cos(π2+x)2sin(π2+x)+sin(3π−x)+sin(3π2+x)+cos(π2+x)
=2cosx+sinx−cosx−sinx=2cosx+sinx−cosx−sinx
=cosx
\(=\dfrac{tan\left(\dfrac{pi}{2}+x\right)\cdot sin\left(-x\right)\cdot cos\left(x-pi\right)}{cos\left(\dfrac{pi}{2}-x\right)\cdot sin\left(x+pi\right)}\)
\(=\dfrac{-cotx\cdot sin\left(-x\right)\cdot\left(-cosx\right)}{sinx\cdot-sinx}\)
\(=\dfrac{cotx\cdot sinx\left(-1\right)\cdot cosx}{-sinx\cdot sinx}=\dfrac{\dfrac{cosx}{sinx}\cdot cosx}{sinx}=\dfrac{cos^2x}{sin^2x}=cot^2x\)
\(=cos\left(x-\frac{\pi}{3}\right)cos\left(x+\frac{\pi}{4}\right)+sin\left(\frac{\pi}{2}-x-\frac{\pi}{6}\right)sin\left(\frac{\pi}{2}-x-\frac{3\pi}{4}\right)\)
\(=cos\left(x-\frac{\pi}{3}\right)cos\left(x+\frac{\pi}{4}\right)+sin\left(\frac{\pi}{3}-x\right)sin\left(-x-\frac{\pi}{4}\right)\)
\(=cos\left(x-\frac{\pi}{3}\right)cos\left(x+\frac{\pi}{4}\right)+sin\left(x-\frac{\pi}{3}\right)sin\left(x+\frac{\pi}{4}\right)\)
\(=cos\left(x-\frac{\pi}{3}-x-\frac{\pi}{4}\right)=cos\left(-\frac{7\pi}{12}\right)=cos\frac{7\pi}{12}=\frac{\sqrt{2}-\sqrt{6}}{4}\)
\(2sin\left(\frac{\pi}{2}+x\right)+sin\left(3\pi-x\right)+sin\left(\frac{3\pi}{2}+x\right)+cos\left(\frac{\pi}{2}+x\right)\)
\(=2cosx+sinx-cosx-sinx\)
\(=cosx\)
\(=cos\left(4\pi+\pi+x\right)+sin\left(4\pi+\frac{\pi}{2}-x\right)-tan\left(\pi+\frac{\pi}{2}+x\right).cot\left(\pi+\frac{\pi}{2}-x\right)\)
\(=cos\left(\pi+x\right)+sin\left(\frac{\pi}{2}-x\right)-tan\left(\frac{\pi}{2}+x\right).cot\left(\frac{\pi}{2}-x\right)\)
\(=-cosx+cosx-\left(-cotx\right).tanx\)
\(=1\)
\(A=cos\left(6\pi+\pi-x\right)+sin\left(2\pi+\frac{\pi}{2}-x\right)+tan^2\left(\pi+\frac{\pi}{2}-x\right)-\frac{1}{sin^2\left(7\pi+\pi+x\right)}\)
\(=cos\left(\pi-x\right)+sin\left(\frac{\pi}{2}-x\right)+tan^2\left(\frac{\pi}{2}-x\right)-\frac{1}{sin^2\left(\pi+x\right)}\)
\(=-cosx+cosx+cot^2x-\frac{1}{sin^2x}\)
\(=cot^2x-\left(1+cot^2x\right)=-1\)
\(sin\left(\frac{\pi}{7}\right)H=sin\left(\frac{\pi}{7}\right)cos\left(\frac{2\pi}{7}\right)+sin\left(\frac{\pi}{7}\right)cos\left(\frac{4\pi}{7}\right)+sin\left(\frac{\pi}{7}\right)cos\left(\frac{6\pi}{7}\right)\)
\(=\frac{1}{2}\left[sin\left(\frac{3\pi}{7}\right)-sin\left(\frac{\pi}{7}\right)+sin\left(\frac{5\pi}{7}\right)-sin\left(\frac{3\pi}{7}\right)+sin\pi-sin\left(\frac{5\pi}{7}\right)\right]\)
\(=-\frac{1}{2}sin\left(\frac{\pi}{7}\right)\)
\(\Rightarrow H=-\frac{1}{2}\)
\(sinA+sinB+sinC=2sin\left(\frac{A+B}{2}\right)cos\left(\frac{A-B}{2}\right)+2sin\left(\frac{C}{2}\right)cos\left(\frac{C}{2}\right)\)
\(=2cos\frac{C}{2}cos\left(\frac{A-B}{2}\right)+2cos\left(\frac{A+B}{2}\right)cos\frac{C}{2}\)
\(=2cos\frac{C}{2}\left[cos\left(\frac{A-B}{2}\right)+cos\left(\frac{A+B}{2}\right)\right]\)
\(=4cos\frac{C}{2}cos\frac{A}{2}cos\frac{B}{2}\)
\(cos\left(2x+\frac{\pi}{6}\right)cos\left(2x-\frac{\pi}{6}\right)=\frac{1}{2}\left(cos4x+cos\frac{\pi}{3}\right)=\frac{1}{2}\left(cos4x+\frac{1}{2}\right)\)
\(sin\left(x+\frac{\pi}{6}\right)sin\left(x-\frac{\pi}{6}\right)=\frac{1}{2}\left(cos\frac{\pi}{3}-cos2x\right)=\frac{1}{2}\left(\frac{1}{2}-cos2x\right)\)
\(\Rightarrow C=\frac{1}{2}sinx.cos4x+\frac{1}{4}sinx+\frac{1}{4}sin3x-\frac{1}{2}sin3x.cos2x\)
\(=\frac{1}{4}sin5x-\frac{1}{4}sin3x+\frac{1}{4}sinx+\frac{1}{4}sin3x-\frac{1}{4}sin5x+\frac{1}{4}sinx\)
\(=\frac{1}{2}sinx\)
\(cosx+cos\left(x+\frac{\pi}{5}\right)+cos\left(x+\frac{9\pi}{5}\right)+cos\left(x+\frac{2\pi}{5}\right)+cos\left(x+\frac{8\pi}{5}\right)+...+cos\left(x+\frac{5\pi}{5}\right)\)
\(=cosx-2cosx.cos\frac{4\pi}{5}-2cosx.cos\frac{3\pi}{5}-2cosx.cos\frac{2\pi}{5}-2cosx.cos\frac{\pi}{5}-cosx\)
\(=-2cosx\left(cos\frac{\pi}{5}+cos\frac{4\pi}{5}+cos\frac{2\pi}{5}+cos\frac{3\pi}{5}\right)\)
\(=-2cosx\left(2cos\frac{\pi}{2}.cos\frac{3\pi}{10}+2cos\frac{\pi}{2}cos\frac{\pi}{10}\right)\)
\(=0\) (do \(cos\frac{\pi}{2}=0\))