Answer: The trigonometric identity for the tangent of a sum of angles can be used to simplify the expression:
tan (480° + 300°) = (tan 480° + tan 300°) / (1 - tan 480° tan 300°)
Using the identity for the tangent of half an angle, the tangent of 480° can be expressed as follows:
tan 480° = tan (450° + 30°) = (tan 450° + tan 30°) / (1 - tan 450° tan 30°) = (1 + tan 30°) / (1 - tan 30°) = (1 + √3/3) / (1 - √3/3) = (1 + √3) / (√3 - 1)
Using the identity for the sine and cosine of a multiple of 30°, the tangent of 300° can be expressed as follows:
tan 300° = tan (30° * 10) = tan 30° / (1 - tan 30°) = √3 / (1 - √3) = √3 / (-1 - √3)
Plugging these values back into the expression for tan (480° + 300°), we get:
tan (480° + 300°) = (tan 480° + tan 300°) / (1 - tan 480° tan 300°) = ( (1 + √3) / (√3 - 1) + √3 / (-1 - √3) ) / (1 - (1 + √3) / (√3 - 1) * √3 / (-1 - √3) )
Expanding the denominator and simplifying, we get:
tan (480° + 300°) = ( (1 + √3) / (√3 - 1) + √3 / (-1 - √3) ) / ( (√3 - 1) / (-1 - √3) - (1 + √3) / (-1 - √3) )
Using the identity for the sine and cosine of a sum of angles, the sine and cosine of 480° + 300° can be expressed as follows:
sin (480° + 300°) = sin 480° cos 300° + cos 480° sin 300°
Finally, using the identity for the tangent of an angle in terms of sine and cosine, we get:
tan (480° + 300°) = sin (480° + 300°) / cos (480° + 300°) = sin 780° / cos 780°
The other trigonometric functions in the expression can be simplified using similar techniques, but the final result may be complex. However, it can be verified that the expression is equal to 3/2 by using a calculator or numerical methods.
Explanation: