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How would you do both these questions? I don't know where to start

For a and b, I try to modulus both sides and then substitute but get back where I started. How would you do these types of questions?

Unsure how to word part a and for part b, I don't know where to start kinda. I tried to substitute and other methods but got nowhere

Just not sure where to start. I wrote out equations where i let a and b be some arbitrary vector and got nowhere.

I've tried to brute force it by distance formula, properties of a parallelogram and finding each point separately and got no success. Would there be a more elegant way of solving this question? Thanks :)

How would you find u-v? I tried this and I got nowhere. I keep ending up with a trig function i cant evaluate. Answer is √5

How would you even determine the graph of the emf v t just based on transformers?? The answer is C

The answer is D. I'm confused on how they arrived at the answer and how actually to determine rotation just based on the charges?

i can do a but i dont know where to start for b. Same with c

im not sure how you would do c given b. I initially thought just base off of inspection that there would be a substitution involving cis or cos theta but its hard for me to tell what to substitute. Or there's just no substirution

I just let tan5x = sin5x/cos5x then use de moivres to get them in terms of their imaginary and real components but from there i cant seem to simplify it to RHS.

Not sure where to start for part b. I tried it a few times starting with RHS and got nowhere and I'm not sure how to introduce cos^3 and sin^4 for LHS

I'm unsure how to apply de mo theorem to these questions because of the +1 in the brackets. How would you apply the theorem in this case?

The answer is A but I'm confused. Isnt velocity always changing? And if A is right wouldn't that make C right aswell? Also, does the angular velocity change or is it constant at these two points and why? Where x is at a higher altitude than y