Question

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

Answer 1

It is not true if (for example) A=3pi/4

Answer 2

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)

Answer 3

tan(3pi/4)=cot(3pi/4)=-1
sec^2(3pi/4)=csc^2(3pi/4)=2
sqrt(2+2)=-1-1 ?

Answer 4

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

Answer 5

In the problem we must add
0<A<pi/2 that's all

Answer 6

Yes, or in more general terms
kπ<A<kπ+π/2 k∈Z