Question
Prove sqrt(Sec^2 A+Cosec^2 A)=TanA+CotA
Answers
Mgraph
It is not true if (for example) A=3pi/4
MathMate
Expand left side into sines and cosines:
sqrt(sec^2A+csc^2A)
=sqrt(1/cos^2A+1/sin^2A)
=sqrt((sin^2A+cos^2A)/(cos^2A sin^2A))
=sqrt(1/(cos^2A sin^2A))
=1/cosA sinA
Similarly expand right hand side:
tanA+cotA
=sinA/cosA + cosA/sinA
=(sin^2A + cos^2A)/(cosA sinA)
=1/(cosA sinA)
sqrt(sec^2A+csc^2A)
=sqrt(1/cos^2A+1/sin^2A)
=sqrt((sin^2A+cos^2A)/(cos^2A sin^2A))
=sqrt(1/(cos^2A sin^2A))
=1/cosA sinA
Similarly expand right hand side:
tanA+cotA
=sinA/cosA + cosA/sinA
=(sin^2A + cos^2A)/(cosA sinA)
=1/(cosA sinA)
Mgraph
tan(3pi/4)=cot(3pi/4)=-1
sec^2(3pi/4)=csc^2(3pi/4)=2
sqrt(2+2)=-1-1 ?
sec^2(3pi/4)=csc^2(3pi/4)=2
sqrt(2+2)=-1-1 ?
MathMate
Here square-root is taken of the square of the product of two functions, and not the numerical values.
To me it is justified to retain the signs of the original functions, namely sin(x) and cos(x) in the square-root.
So if we evaluate the functions after taking square-root,
LHS=1/(cos(3π/4)sin(3π/3)=-2
and
RHS=-2 as you have calculated.
As a compromise, we can say that the identity should read:
(Sec^2 A+Cosec^2 A)=(TanA+CotA)²
To me it is justified to retain the signs of the original functions, namely sin(x) and cos(x) in the square-root.
So if we evaluate the functions after taking square-root,
LHS=1/(cos(3π/4)sin(3π/3)=-2
and
RHS=-2 as you have calculated.
As a compromise, we can say that the identity should read:
(Sec^2 A+Cosec^2 A)=(TanA+CotA)²
Mgraph
In the problem we must add
0<A<pi/2 that's all
0<A<pi/2 that's all
MathMate
Yes, or in more general terms
kπ<A<kπ+π/2 k∈Z
kπ<A<kπ+π/2 k∈Z