cos^(-1) x + sin^(-1) x

Question

cos^(-1) x + sin^(-1) x

Answer 1

If by cos^(-1)x you mean arccos(x) and by sin^(-1)x you mean arcsin(x),  then the sum is pi/2

 

In other words,   the angles  arccos(x) and arcsin(x) are complimentary. 

Answer 2

usually cos^-1x refers to the inverse function which is arcos x whereas sec x is called the reciprocal function.

 

So perhaps this is intended to be arcos x + arcsin x

Answer 3

Usually, cos^-1 x means the inverse function of cosine. Consider y = cos x. normally you put in an angle measurement for x and get some decimal for y. The inverse function does the exact opposite, you put in some decimal and you get out some angle measurement. for example,

if we consider only positive acute angles

cos^-1 (1/2) = 60 degrees

cos^-1 (.76604) = 40 degrees

 

If the problem intended you to raise cosine x to the -1 power, they would write (cos x)^-1 which is indeed the definition of sec x.

 

As to an answer to your problem, if you draw out a right triangle, with legs AC and BC, the hypotenuse would be AB.

Look at cos^-1 x as cos^-1 (x/1) and let's say that is angle A. That means the adjacent leg, AC = x and the hypotenuse AB = 1.

If you drew this diagram cos A =x/1 or cos^-1 (x/1) = A. Hope you're following so far.

 

If you look at angle B you may notice that the sin B = x/1 so sin^-1 (x/1) = B.

If we put these 2 facts together, cos^-1 (x/1) + sin^-1 (x/1) = A + B.

 

Look at your triangle and you should notice these angles must add to 90 since the third angle is the right angle and a triangles angle measurement add to 180. so A + B = 90, so cos^-1 x + sin^-1 x =90.

 

Hope you followed all this.

Answer 4

The inverse of sin x is csc x and the inverse of cos x is sec x, so the answer is sec x + csc x