Question

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Verify the identity. tan x plus pi divided by two = -cot x

Answer 1

`\tan\left(x+(\pi)/(2)\right)=-\cot(x)\\\\L=(\sin\left(x+(\pi)/(2)\right))/(\cos\left(x+(\pi)/(2)\right))=(\sin(x)\cos(\pi)/(2)+\sin(\pi)/(2)\cos(x))/(\cos(x)\cos(\pi)/(2)-\sin(x)\sin(\pi)/(2))\\\\=(\sin(x)\cdot0+1\cdot\cos(x))/(\cos(x)\cdot0-\sin(x)\cdot1)=(\cos(x))/(-\sin(x))=-\cot(x)=R\\\\\\Used:\\\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)\\\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)\\\cot(x)=(\cos(x))/(\sin(x))\\\sin(\pi)/(2)=1\\\cos(\pi)/(2)=0`