The value of sin^2 30 - cos^2 30

To find the value of sin^2 30 - cos^2 30, we can start by using the trigonometric identities for sine and cosine.

Sine Identity:
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.

Cosine Identity:
The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

Step 1: Finding the values of sin 30 and cos 30
We know that sin 30 = 1/2 and cos 30 = √3/2.

Step 2: Evaluating sin^2 30 - cos^2 30
Using the values we found in step 1, we can substitute them into the expression sin^2 30 - cos^2 30.

sin^2 30 - cos^2 30 = (1/2)^2 - (√3/2)^2

Simplifying further, we have:

sin^2 30 - cos^2 30 = 1/4 - 3/4

Now, subtracting the fractions, we get:

sin^2 30 - cos^2 30 = -2/4

Step 3: Simplifying the fraction
The fraction -2/4 can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2.

-2/4 = -1/2

Step 4: Final Answer
Therefore, sin^2 30 - cos^2 30 = -1/2.

Summary:
The value of sin^2 30 - cos^2 30 is -1/2. This can be found by first identifying the values of sin 30 and cos 30, substituting them into the expression, and then simplifying the resulting fraction.