Question

Verify the identity.

tan x plus pi divided by two = -cot x

Answer 1

recall that sin(π/2) = 1, and cos(π/2) = 0,


`\bf \textit{Sum and Difference Identities} \\\\ sin(\alpha + \beta)=sin(\alpha)cos(\beta) + cos(\alpha)sin(\beta) \\\\ sin(\alpha - \beta)=sin(\alpha)cos(\beta)- cos(\alpha)sin(\beta) \\\\ cos(\alpha + \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta) \\\\ cos(\alpha - \beta)= cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta)\\\\ -------------------------------\\\\`


`\bf tan\left(x+(\pi )/(2) \right)=-cot(x)\\\\ -------------------------------\\\\ tan\left(x+(\pi )/(2) \right)\implies \cfrac{sin\left(x+(\pi )/(2) \right)}{cos\left(x+(\pi )/(2) \right)} \\\\\\ \cfrac{sin(x)cos\left((\pi )/(2) \right)+cos(x)sin\left((\pi )/(2) \right)}{cos(x)cos\left((\pi )/(2) \right)-sin(x)sin\left((\pi )/(2) \right)}\implies \cfrac{sin(x)\cdot 0~~+~~cos(x)\cdot 1}{cos(x)\cdot 0~~-~~sin(x)\cdot 1} \\\\\\ \cfrac{cos(x)}{-sin(x)}\implies -cot(x)`