Given that cos 75 = X, show that cos 105 = −X

Step-by-step explanation:

cos(90) = 0

around this point the cos function "mirrors" with opposite signs. cos(<90) is positive and cos(>90) is negative.

but |cos(90-a)| = |cos(90+a)| for 0 <= a <= 90

75 = 90 - 15

105 = 90 + 15

so, a = 15

and because of

|cos(90-a)| = |cos(90+a)| for 0 <= a <= 90

cos (90-15) = cos(75) = -cos(90+15) = -cos(105)