Part I .
sinh^4A+cosh^4A=(cosh4A+3)/4
Use the formulas:
cosh 2A=2cosh^2A-1 -->Solve for cosh^2A
cosh 2A=1+2sinh^2A --->Solve for sinh^2A
Left Hand Side:
sinh^4A+cosh^4A= (sinh^2 A)^2 +(cosh^2A)^2
=((cosh2A-1)/2)^2+ ((cosh2A+1)/2)^2 -->FOIL
=(cosh^2 2A -2cosh2A+1)/4 +(cosh^2 2A+2cosh2A+1)/4
=(cosh^2 2A -2cosh2A+1 +cosh^2 2A+2cosh2A+1)/4
=(cosh^2 2A -cancel(2cosh2A)+1 +cosh^2 2A+cancel(2cosh2A)+1)/4
=(cosh^2 2A +1 +cosh^2 2A+1)/4
=(2cosh^2 2A +2)/4
Note: cosh 4A = 2cosh^2 2A -1 :. cosh 4A+1 = 2cosh^2 2A
=(cosh 4A+1 +2)/4
=(cosh 4A+3)/4
:.= Right Hand Side
Part II
cosh^4A-sinh^4A=cosh 2A
Use the properties :
cosh^2A-sinh^2A =1 and cosh 2A=1+2sinh^2A
Left Hand Side :
cosh^4A-sinh^4A=(cosh^2A+sinh^2A)(cosh^2A-sinh^2A)
=(cosh^2A+sinh^2A)*1
=cosh^2A+sinh^2A
=1+sinh^2A+sinh^2A
=1+2sinh^2A
=cosh 2A
:.= Right Hand Side
