(sec 2 (20)−1)(1−csc 2 (20)) is equal to:

To solve the expression (sec^2(20) - 1)(1 - csc^2(20)), we can break it down into smaller steps:

Step 1: Evaluate sec^2(20) To find sec^2(20), we need to calculate the square of the secant of 20 degrees. The secant of an angle is equal to 1 divided by the cosine of that angle. So, sec(20) = 1/cos(20). Therefore, sec^2(20) = (1/cos(20))^2 = 1/cos^2(20).

Step 2: Evaluate csc^2(20) Similarly, to find csc^2(20), we need to calculate the square of the cosecant of 20 degrees. The cosecant of an angle is equal to 1 divided by the sine of that angle. So, csc(20) = 1/sin(20). Therefore, csc^2(20) = (1/sin(20))^2 = 1/sin^2(20).

Step 3: Substitute the values into the expression Now that we have the values for sec^2(20) and csc^2(20), we can substitute them into the original expression: (sec^2(20) - 1)(1 - csc^2(20)) = (1/cos^2(20) - 1)(1 - 1/sin^2(20)).

Step 4: Simplify the expression To simplify the expression further, we can use the identity: 1 - cos^2(20) = sin^2(20). Applying this identity, we get: (1/cos^2(20) - 1)(1 - 1/sin^2(20)) = (1/cos^2(20) - cos^2(20)/cos^2(20))(1 - 1/sin^2(20)).

Simplifying the expression inside the parentheses, we have: (1 - cos^2(20))(1 - 1/sin^2(20)) = sin^2(20)/cos^2(20) * sin^2(20)/sin^2(20).

Step 5: Simplify the expression further Multiplying the fractions, we get: sin^4(20)/cos^2(20) * 1/sin^2(20).

Simplifying the expression, we have: sin^4(20)/cos^2(20) * 1/sin^2(20) = sin^4(20)/(cos^2(20) * sin^2(20)).

Step 6: Simplify the expression to its final form Using the trigonometric identity: sin^2(20) + cos^2(20) = 1, we can rewrite the expression as: sin^4(20)/(cos^2(20) * sin^2(20)) = sin^4(20)/(1 - sin^2(20)).

And that is the final simplified form of the expression (sec^2(20) - 1)(1 - csc^2(20)).