State whether the following is 'True' or 'False' and justify your answer:
If $\cos A + \cos^2 A = 1$,then $\sin^2 A + \sin^4 A = 1$.
If $\cos A + \cos^2 A = 1$,then $\sin^2 A + \sin^4 A = 1$.
(A) True.
Given that $\cos A + \cos^2 A = 1$.
Rearranging the terms,we get $\cos A = 1 - \cos^2 A$.
Using the identity $\sin^2 A + \cos^2 A = 1$,we know that $1 - \cos^2 A = \sin^2 A$.
Therefore,$\cos A = \sin^2 A$.
Squaring both sides,we get $\cos^2 A = (\sin^2 A)^2 = \sin^4 A$.
We know that $\cos^2 A = 1 - \sin^2 A$.
Substituting this into the equation $\cos^2 A = \sin^4 A$,we get $1 - \sin^2 A = \sin^4 A$.
Rearranging the terms,we get $\sin^2 A + \sin^4 A = 1$.
Given that $\cos A + \cos^2 A = 1$.
Rearranging the terms,we get $\cos A = 1 - \cos^2 A$.
Using the identity $\sin^2 A + \cos^2 A = 1$,we know that $1 - \cos^2 A = \sin^2 A$.
Therefore,$\cos A = \sin^2 A$.
Squaring both sides,we get $\cos^2 A = (\sin^2 A)^2 = \sin^4 A$.
We know that $\cos^2 A = 1 - \sin^2 A$.
Substituting this into the equation $\cos^2 A = \sin^4 A$,we get $1 - \sin^2 A = \sin^4 A$.
Rearranging the terms,we get $\sin^2 A + \sin^4 A = 1$.
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