Triangle ABC is a isosceles right-angled triangle.A = C = 45 degrees.Therefore, AB = BC = Sq.root(2)

Area of triangle ABC = 1/2 * Sq.root(2) * Sq.root(2)
= 1/2 * 2
= 1

In Triangle DEF , angle F= 30 degrees.

Sin 30 = 1/2 =DE/2

DE = 1

Also, Cos 30 = Sq.root(3)/2 = EF/2

EF = Sq.root(3) = 1.732

This implies area of triangle DEF = 1/2 * 1 * 1.732
= .866

Clearly,Area of ABC is greater than Area of DEF.

Do they ask questions based on sin and cos formulas in GRE ?

Trigonometry is not tested on GRE or GMAT.
This means there is always another way to do the question..

Here we have two types of triangles..
45-45-90 sides are 1:1:√2 or √2:√2:2..Area = √2*√2*1/2=1=A
30-60-90 sides are 1:√3:2....Area = 1*√3*1/2=√3/2=1.74/2=0.87=B

Thus A>B
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html