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
