cos−1x=sin−1xWe know that the ranges of the inverse trigonometric functions limit the values of
x. Specifically, for both
cos−1x and
sin−1x,x must be in the range
[−1,1].
Let's denote the common value of
cos−1x and
sin−1x by
θ. Therefore, we have:
θ=cos−1x=sin−1xFrom the properties of inverse trigonometric functions, we know:
cosθ=xsinθ=xWe also know from trigonometric identities that:
cos2θ+sin2θ=1Substituting
x into the identity, we get:
x2+x2=12x2=1x2=x=±Since
x must fall within the range
[−1,1], both positive and negative values are valid within this context. However, given the problem's options, the correct answer aligns with only the positive value provided in the options list.