A level further maths arc-trig

What is the maximum value of (arcsinx)(arccos(x)?
Find the answer:
a) by differentiating
b) by considering the value of arcsinx + arccosx
Can anyone help me with this?
For part a) I got the equation arcsinx = arccosx but I'm not sure how to get a value for x from that.
I'm not entirely sure how to start part b)

Is it
arcsin(x) *arccos(x)
For a) it would be product rule and standard derivatives of arc trig functions. If youve done that and got to arcsin = arccos, then its pretty much write down / use the cos - sin definition/identity.
b) looks like fundamental defn/identity of cos and - sin again and am/gm inequality (binomial) approach done from scratch but it would be good to post a pic of the question and what youve done.

Looks about right for a) as well as the sketch. Can you just reason/guess about what the value of x and hence the value of arcsin(x)*arccos(x)?
If not, the identity/definition you want is
cos(x) = sin(pi/2 - x)
as cos stands for COmplementary angle Sin as should be obvious if you just draw a right triangle. See if you can map that to the inverse trig functions (should be straight forward, but its worth you being "happy" about it)

Would the value of x be pi/4? Do I substitute this back into the question to find the maximum value? Sorry but I'm not really sure what you mean by mapping a right triangle to the inverse trig functions 😭 is this still for part a) or part b)?

Thats correct for the answer, though if youre not happy about the cos-sin thing, its worth doing the pi/4 analysis "rigorously". The identity is pretty much by defn of a right triangle so at the top of
https://en.wikipedia.org/wiki/Right_triangle
cos(A) = sin(pi/2-A) = sin(B)
as both refer to base/hyp = b/c. B is the complementary angle to A.
so start with
arccos(x) = arcsin(x)
and take cos of both sides and reason about the right. Or take sin of both sides and reason about the left. The complementary part should become clear and how you get pi/4 or 1/sqrt(2)

Ohh yes I understand that part thank you so much!! Do you have any ideas of how I could start part b)?

This is what I've done so far for part b) but I'm still not sure about how to get rid of the arcsin^2(x) and arccos^2(x)

This is what I've done so far for part b) but I'm still not sure about how to get rid of the arcsin^2(x) and arccos^2(x)

After a bit of thought (always helps), another way to do part b) would be once you have the result for
arcsin + arccos
would be to write one in terms of the other and simply sub back into the original expression to maximise. That will give the result.
Doing am-gm is overkill for the question, and I thought that originally but got sidetracked by it. Its a valid way but more complex than the question needs..

What is the maximum value of (arcsinx)(arccos(x)?
Find the answer:
a) by differentiating
b) by considering the value of arcsinx + arccosx
Can anyone help me with this?
For part a) I got the equation arcsinx = arccosx but I'm not sure how to get a value for x from that.
I'm not entirely sure how to start part b)