If sinx = 12/13, where 0°<x<90°, find the value of (1-cos²x)/(1-sin²30°)

Here are the solutions to the next set of problems. 6. Given that 30^ = 60^ = sqrt(3)2 and 30^ = 60^ = (1)/(2), evaluate (( 60^ - 1))/((1 - 30^)). Step 1: Find the values of 60^ and 30^. 60^ = ( 60^)/( 60^) = sqrt(3)2(1)/(2) = sqrt(3) 30^ = ( 30^)/( 30^) = (1)/(2)sqrt(3)2 = (1)/(sqrt(3)) Step 2: Substitute these values into the expression. (sqrt(3) - 1)(1 - (1)/(sqrt(3))) Step 3: Simplify the denominator. 1 - (1)/(sqrt(3)) = sqrt(3)sqrt(3) - (1)/(sqrt(3)) = sqrt(3) - 1sqrt(3) Step 4: Substitute the simplified denominator back into the main expression and simplify. sqrt(3) - 1sqrt(3) - 1sqrt(3) = (sqrt(3) - 1) × sqrt(3)sqrt(3) - 1 = sqrt(3) The value is sqrt(3). 7. If x = (12)/(13) where 0^ < x < 90^, find the value of (1 - ^2 x). Step 1: Use the trigonometric identity ^2 x + ^2 x = 1. Rearranging the identity, we get 1 - ^2 x = ^2 x. Step 2: Substitute the given value of x into the identity. 1 - ^2 x = ^2 x = ((12)/(13))^2 Step 3: Calculate the square. ((12)/(13))^2 = (12^2)/(13^2) = (144)/(169) The value is (144)/(169). 8. Without using tables or calculators, calculate the value of ((1 + ^2 30^))/((1 - ^2 30^)). Step 1: Recall the values of 30^ and 30^. 30^ = sqrt(3)2 30^ = (1)/(2) Step 2: Substitute these values into the expression. (1 + (sqrt(3)2)^2)(1 - ((1)/(2))^2) Step 3: Calculate the squares. (sqrt(3)2)^2 = (3)/(4) ((1)/(2))^2 = (1)/(4) Step 4: Substitute the squared values back into the expression and simplify the numerator and denominator. ((1 + 3)/(4))(1 - (1)/(4)) = ((4)/(4) + (3)/(4))((4)/(4) - (1)/(4)) = (7)/(4)(3)/(4) Step 5: Divide the numerator by the denominator. (7)/(4)(3)/(4) = (7)/(4) × (4)/(3) = (7)/(3) The value is (7)/(3). That's 2 down. 3 left today — send the next one.