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fenix001 [56]
3 years ago
6

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

Mathematics
1 answer:
kogti [31]3 years ago
7 0
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+\frac{\pi }{2}  \right)=-cot(x)\\\\
-------------------------------\\\\
tan\left(x+\frac{\pi }{2}  \right)\implies \cfrac{sin\left(x+\frac{\pi }{2}  \right)}{cos\left(x+\frac{\pi }{2}  \right)}
\\\\\\
\cfrac{sin(x)cos\left(\frac{\pi }{2}  \right)+cos(x)sin\left(\frac{\pi }{2}  \right)}{cos(x)cos\left(\frac{\pi }{2}  \right)-sin(x)sin\left(\frac{\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)
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