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[Astudious]

Is there any way I can reduce or simplify expressions like cos(sin^-1(x)), sin(cos^-1(x)), cos(tan^-1(x)), tan(cot^-1(x)) etc.? (I refer to the arc functions, i.e. inverses, by the superscript -1, not the reciprocals)

 

[Svein]

Hint: sin(sin^-1(x)) = x and cos(w) = sin(w+π/2).

 

[abbas_majidi]

You can use this relation:
ArcSin((1-x^2)^0.5)=ArcCos(x)

 

[Mentallic]

Is there any way I can reduce or simplify expressions like cos(sin^-1(x))

Draw a right triangle with an angle $\theta$, opposite side length of x and hypotenuse length 1. Now from this triangle, by definition,

$$\sin\theta = x$$
hence
$$\theta = \sin^{-1}x$$

So then what is $\cos\left(\sin^{-1}x\right)$, or more simply, $\cos\theta$ ?