Prove that following statements.
For any real numbers x and y, if x = y, then `x^2 = y^2`.

For any real number of x and y, if cos x=cos y then:

State whether the following statements are true or false. (x -y) (x +y) = x ^(2) + y ^(2)

Find real numbers x and y such that 3x +2iy-ix+5y =7 +5i .

Check whether the following pair of statements are negation of each other. Give reasons for your answer. x + y= y + x is true for every real numbers x and y. There exists real numbers x and y for which x + y = y + x .

Restate the following statements with appropriate conditions, so that they become true. : If x^2 = y^2 , then x = y .

Check whether the following pairs of statements are negation of each other. Give reasons for your answer. There exists real number x and y for which x+y=y+x There exists Real number x and y for which x+y=xy

Let [a] denotes the larger integer not exceeding the real number a if x and y satisfy the equations y=2[x]+3 and y=3[x-2[ simultaneously determine [x+y]

Prove that the following relation R in Z of integers is an equivalence relation : R = [ (x, y) : 2x - 2y is an integer}

Prove the following : (sinx +siny)/(cos x+cos y)= tan frac (x+y)(2) .

Check whether the following pairs of statements are negation of each other. Give reasons for your answer. x+y=y+x is true for every real number x and y. There exists real number x and y for which x +y = y+x