Question

Sin(x-pi/2)=-cosx
Prove the identity

Answer 1

let's bear in mind 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) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ sin\left(x-(\pi )/(2) \right)\implies sin(x)cos\left( (\pi )/(2) \right)-cos(x)sin\left( (\pi )/(2) \right) \\\\\\ sin(x)[0]-cos(x)[1]\implies -cos(x)`