Asked by Vijaya

Prove sqrt(Sec^2 A+Cosec^2 A)=TanA+CotA

Answers

Answered by Mgraph
It is not true if (for example) A=3pi/4
Answered by 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)
Answered by Mgraph
tan(3pi/4)=cot(3pi/4)=-1
sec^2(3pi/4)=csc^2(3pi/4)=2
sqrt(2+2)=-1-1 ?
Answered by 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

Answered by Mgraph
In the problem we must add
0<A<pi/2 that's all
Answered by MathMate
Yes, or in more general terms
kπ<A<kπ+π/2 k∈Z
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