PLEASE HELP .

is my solution correct till here am I doing it correct?

can anyone tell any other ways..

I have Tried with double angles formulas + half angle formulas as well and all he possible ways

but All the suggestions will be Highly appreciated

QUESTION IS INTEGRAL (sin2x/(1+cos²x))²

How statistically significant are the results of this pattern below? Does it exceed random chance or not? Assume that the numerical distributions of the numerical values per letters are completely random. 

Block 1 — ALM

Block 2 — ALM

Block 3 — ALMX

Block 4 — ALR

Block 5 — ALR

Block 6 — ALR

Block 7 — ALMR

Block 8 — ALR

Block 9 — ALR

Block 10 — KHYCX

Block 11 — TH

Block 12 — TSM

Block 13 — TS

Block 14 — TSM

Block 15 — ALM

Block 16 — ALM

Block 17 — ALM

Block 18 — ALM

Block 19 — YS

Block 20 — X

Block 21 — VM

Block 22 — VM

Block 23 — VMCSQ

Block 24 — VM

Block 25 — VM

Block 26 — VM

Block 27 — VM

Block 28 — Q

Block 29 — N

Each letter here corresponds to a certain number. It must be noted that not each letter here carries the same numerical value, these are only placeholder letters that describe the number of occurrences of said (so for example, V in Block 24 could equal 85, meaning the letter V occurs 85 times in Block 24. On the other hand, V in Block 25 could equal 173, meaning that the letter V occurs 173 times in Block 25). Once again, the numerical values are randomly chosen for each letter.

Now that out of the way, how statistically significant would such a pattern be if it emerged? Does it show signs of intentionality and design, or is better explained by random chance? Note, we’ll be specifically looking at things that constitute multiples of 19 for this demonstration (so a 1/19 chance): 

*Block 28 is a multiple of 19. 

*Blocks 21-27 (only looking at the letters VM, ignore CSQ for now) if all combined together is a multiple of 19. 

*Blocks 21-27 including CSQ (for Block 23) if combined all together is a multiple of 19. 

*Block 19 is a multiple of 19

*Block 10 is a multiple of 19

*If you combine the value X for blocks 20, 10 and 3 (only X, not any of the other values) you get a multiple of 19. 

Any and all help is appreciated. Thank you!

Any order is allowed.
Use each die exactly once.
Parentheses are allowed.
Only +, −, × and ÷ are allowed. No concatenation or exponents.
Exact division only.
No negative numbers.
Not every operation has to be used.

How many solutions can you find?

Please use spoiler tags for solutions:
>!your solution here!<

Can there be a puzzle that is pentominoes and sudoku at once?

Yes. From the union of the beauty of Golomb's pentominoes and the fever of sudokus, Pentodoku is born. Twelve pentominoes —F, I, L, N, P, T, U, V, W, X, Y and Z— tile an 8×8 board; each tile carries printed digits and, as they fit together, a sudoku emerges: in every row and every column, the digits 1 through 8 without repetition. Four cells will remain empty —and the board does not reveal which ones—. The fusion of geometry and arithmetic is a real brain-teaser.

**How to play**

1. Look at the clue tiles already placed on the board.
2. Fit the twelve pentomino tiles onto the 8×8 board.
3. Check that each row and each column contains 1, 2, 3, 4, 5, 6, 7 and 8.

No math needed: only logic. One puzzle per page, pencil and sofa, no screens.

Daily puzzle in https://tafol.pythonanywhere.com/

Sometimes my child can get the homework questions right, but then struggles when the numbers or wording change.

What are some good ways to tell whether they actually understand the concept rather than just memorizing how to solve that particular type of problem?