# Trigonometric Equations

1. Apr 3, 2015

### 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)

2. Apr 3, 2015

### Svein

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

3. Apr 3, 2015

### abbas_majidi

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

4. Apr 3, 2015

### Mentallic

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, $\cos\theta$ ?