Write cos(arcsin3x +arccos x) as an algebraic expression of x that does not involve trigonometric functions.

Step-by-step explanation:

Since we have to analyze the cosine of an addition of angles, let's recall that cos of an addition is:

Let's also use those Greek letters and to represent the inverse functions we are dealing with:

and therefore,

and therefore

Now use these identities to re-write the equation for the cos of an addition given above:

Now recall that the sine of an angle can be written in terms of the cos of the same angle via the Pythagorean identity:

In our case, and with our particular values, this gives:

Similarly we can write the cos of an angle in terms of the sine of the same angle:

In ours case, using our values, this gives:

So replacing these in the cos of an addition of angles formula, we obtain that the given expression equals:

check the picture below.

well, the angle at the vertex d is clearly not a right-angle, let's check for the one at the vertex f which does look like,   however if it's indeed, that means ef is perpendicular to df, in which case their slopes will be negative reciprocal of each other.

so the negative reciprocal of the slope of ef is 1, however the slope of df is not 1, is 1/2, so nope, they're not perpendicular lines.

b, i believe, but not sure.