`1-2ab - (a ^(2) +b ^(2)) = ? `

Factorise : 1 + 2 ab - (a ^(2) + b ^(2))

The value of the determinant |{:(1+ a^(2) - b^(2),2 ab , - 2b),(2ab, 1 - a^(2) + b^(2), 2a),(2b , -2a , 1-a^(2) - b^(2)):}| is equal to

By using properties of determinants , show that : {:[( 1+a^(2) -b^(2) ,2ab , -2b),( 2ab, 1-a^(2) +b^(2) , 2a),( 2b, -2a, 1-a^(2) -b^(2)) ]:}=( 1+a^(2) +b^(2)) ^(3)

Show that the tangent of an angle between the lines (x)/(a) + (y)/( b) = 1 and (x)/( a) - (y)/( b) = 1 is (2 ab)/( a^(2) - b^(2) ) .

If cos alpha+ cos beta= a and sin alpha+ sin beta =b , then show that sin 2 alpha+ sin 2 beta =2 ab(1- (2)/(a^(2)+ b^(2))) .

If [ sqrt (a ^(2) + b^(2) + ab)] + [ sqrt (a^(2) + b^(2) - ab)] = 1 then what is the value of (1 - a^(2)) (1 - b^(2)) ?