Concept:Formula used:cos 2? = 2cos2? - 1cos(2? - ?) = cos ? cos(? + ?) = - cos ? (a - b)2 = a2 + b2 - 2ab(a + b)2 = a2 +b2 + 2ab Calculation:Let the required value is y.? y = \(\cos^4\frac{7?}{8}+\cos^4\frac{5?}{8}\)? y = \((\cos^2\frac{7?}{8})^2+(\cos^2\frac{5?}{8})^2\)Using the formula discussed above? y = \((\frac{cos \frac{2\times7? }{8}+1}{2})^2+(\frac{cos \frac{2\times5? }{8}+1}{2})^2\)? y = \((\frac{\cos(2\pi- \frac{?}{4})+1}{2})^2+(\frac{\cos(\pi+\frac{?}{4})+1}{2})^2\)Using the formula (2) & (3)? y = \((\frac{\cos \frac{?}{4}+1}{2})^2+(\frac{\cos\frac{?}{4}-1}{2})^2\)Sicne, cos(?/4) = 1/?2? y = \((\frac{\ \frac{1}{\sqrt 2}+1}{2})^2+(\frac{1-\frac{1}{\sqrt 2}}{2})^2\)? y = \((\frac{\sqrt 2 +1}{2\sqrt2})^2+(\frac{\sqrt 2-1 }{2\sqrt2})^2\)Using the formula (4) & (5)? y = \(\frac{ (2 + 1+2\sqrt 2)+(2 + 1 - 2\sqrt 2)}{8}\)? y = 6/8 = 3/4? \(\cos^4\frac{7?}{8}+\cos^4\frac{5?}{8}\) = 3/4