How do you verify tan(x+π2)=−cotx?

1 Answer
Jorge G. Pires
Feb 16, 2016

By applying the basic trigonometric relations. see below

Explanation:

head

Key-relation 1. tanx=sinxcosx

Key-relation 2. cotx=1tanx=cosxsinx

Key-relation 3. cos(a+b)=cosa⋅cosb−sina⋅sinb

Key-relation 4. sin(a+b)=cosa⋅sinb+sina⋅cosb

Import results

Important result 1. cos(π2)=0
Important result 1. sin(π2)=1

Gathering

By using all the knowledge left so far and some mathematical tricks, we have:

sin(x+(π2))=cosx⋅sin(π2)+sinx⋅cos(π2)
sin(x+(π2))=cosx

Further:

cos(x+(π2))=cosx⋅cos(π2)−sinx⋅sin(π2)

cos(x+(π2))=−sinx

Finally:

tan(x+(π2))=sin(x+(π2))cos(x+(π2))

tan(x+(π2))=−cosxsinx=cotx

End of the proof!

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