prove it please
answer only if you know​

Part (c)

We'll use this identity

[tex]\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)\\\\[/tex]

to say

[tex]\sin(A+45) = \sin(A)\cos(45) + \cos(A)\sin(45)\\\\\sin(A+45) = \sin(A)\frac{\sqrt{2}}{2} + \cos(A)\frac{\sqrt{2}}{2}\\\\\sin(A+45) = \frac{\sqrt{2}}{2}(\sin(A)+\cos(A))\\\\[/tex]

Similarly,

[tex]\sin(A-45) = \sin(A + (-45))\\\\\sin(A-45) = \sin(A)\cos(-45) + \cos(A)\sin(-45)\\\\\sin(A-45) = \sin(A)\cos(45) - \cos(A)\sin(45)\\\\\sin(A-45) = \sin(A)\frac{\sqrt{2}}{2} - \cos(A)\frac{\sqrt{2}}{2}\\\\\sin(A-45) = \frac{\sqrt{2}}{2}(\sin(A)-\cos(A))\\\\[/tex]

-------------------------

The key takeaways here are that

[tex]\sin(A+45) = \frac{\sqrt{2}}{2}(\sin(A)+\cos(A))\\\\\sin(A-45) = \frac{\sqrt{2}}{2}(\sin(A)-\cos(A))\\\\[/tex]

Therefore,

[tex]2\sin(A+45)*\sin(A-45) = 2*\frac{\sqrt{2}}{2}(\sin(A)+\cos(A))*\frac{\sqrt{2}}{2}(\sin(A)-\cos(A))\\\\2\sin(A+45)*\sin(A-45) = 2*\left(\frac{\sqrt{2}}{2}\right)^2\left(\sin^2(A)-\cos^2(A)\right)\\\\2\sin(A+45)*\sin(A-45) = 2*\frac{2}{4}\left(\sin^2(A)-\cos^2(A)\right)\\\\2\sin(A+45)*\sin(A-45) = \sin^2(A)-\cos^2(A)\\\\[/tex]

The identity is confirmed.

==========================================================

Part (d)

[tex]\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)\\\\\sin(45+A) = \sin(45)\cos(A) + \cos(45)\sin(A)\\\\\sin(45+A) = \frac{\sqrt{2}}{2}\cos(A) + \frac{\sqrt{2}}{2}\sin(A)\\\\\sin(45+A) = \frac{\sqrt{2}}{2}(\cos(A)+\sin(A))\\\\[/tex]

Similarly,

[tex]\sin(45-A) = \sin(45 + (-A))\\\\\sin(45-A) = \sin(45)\cos(-A) + \cos(45)\sin(-A)\\\\\sin(45-A) = \sin(45)\cos(A) - \cos(45)\sin(A)\\\\\sin(45-A) = \frac{\sqrt{2}}{2}\cos(A) - \frac{\sqrt{2}}{2}\sin(A)\\\\\sin(45-A) = \frac{\sqrt{2}}{2}(\cos(A)-\sin(A))\\\\[/tex]

-----------------

We'll square each equation

[tex]\sin(45+A) = \frac{\sqrt{2}}{2}(\cos(A)+\sin(A))\\\\\sin^2(45+A) = \left(\frac{\sqrt{2}}{2}(\cos(A)+\sin(A))\right)^2\\\\\sin^2(45+A) = \frac{1}{2}\left(\cos^2(A)+2\sin(A)\cos(A)+\sin^2(A)\right)\\\\\sin^2(45+A) = \frac{1}{2}\cos^2(A)+\frac{1}{2}*2\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\\sin^2(45+A) = \frac{1}{2}\cos^2(A)+\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\[/tex]

and

[tex]\sin(45-A) = \frac{\sqrt{2}}{2}(\cos(A)-\sin(A))\\\\\sin^2(45-A) = \left(\frac{\sqrt{2}}{2}(\cos(A)-\sin(A))\right)^2\\\\\sin^2(45-A) = \frac{1}{2}\left(\cos^2(A)-2\sin(A)\cos(A)+\sin^2(A)\right)\\\\\sin^2(45-A) = \frac{1}{2}\cos^2(A)-\frac{1}{2}*2\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\\sin^2(45-A) = \frac{1}{2}\cos^2(A)-\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\[/tex]

--------------------

Let's compare the results we got.

[tex]\sin^2(45+A) = \frac{1}{2}\cos^2(A)+\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\\sin^2(45-A) = \frac{1}{2}\cos^2(A)-\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\[/tex]

Now if we add the terms straight down, we end up with [tex]\sin^2(45+A)+\sin^2(45-A)[/tex] on the left side

As for the right side, the sin(A)cos(A) terms cancel out since they add to 0.

Also note how [tex]\frac{1}{2}\cos^2(A)+\frac{1}{2}\cos^2(A) = \cos^2(A)[/tex] and similarly for the sin^2 terms as well.

The right hand side becomes [tex]\cos^2(A)+\sin^2(A)[/tex] but that's always equal to 1 (pythagorean trig identity)

This confirms that [tex]\sin^2(45+A)+\sin^2(45-A) = 1[/tex] is an identity

1. observe the following tables and find if x and y are directly proportional:

Give answer plzz​

ANSWERS THIS PLZ PLZ PLZ!!!!!

need help please somebodt

(b) A private club grew by 7 members each week for 42 weeks. What was the total change in the
club's size?
members please help

Pairs of shoes are manufactured for $24. They are sold to a warehouse with a mark-up of 15%. The warehouse sells the shoes to a distributor after charging a holding fee of $10 per pair. The distributor sells them to ‘Fine Shoes’ for a percentage profit of 12%. The store then marks them up by 30%.


a. Determine the price of a pair of shoes if you buy it from one of the ‘Fine Shoes’ stores (round to the nearest 5 cents). Show working out.

can someone help plz

Henry decided he was going to give away his Pokemon cards. He divided them equally into 4 piles. Each person got 37 cards. How many cards did Henry have in the beginning?

pls help i need the answer and the equation

Please and thanks it’s due soon!!!

Plz help!!!!!!!!!!!!!!!!!!!!!

what is the value of x?

1.If a shirt costs 60 dollars, it's marked down to 20% off, what is the amount of the discount?

how many grams are in 4.0 moles of H2O

please help with this

classify 1/3 if it is rational, irrational, integer, whole, or natural​

Need help te slim is designing a cover for a book he wants to have a rectangler grid

Help me with this problem please.

Which statements are true about the graph of f(x) = x sin 1/x?

i’ll mark brainliest or whatever

2.)
Find the perimeter of an isosceles triangle whose base is 16 cm and whose height is 15 cm.

The equation of a line is y= -3x - 18. What are the coordinates of the point on the
line when the value of yis 3?

Enter the correct answer in the box.
The formula for centripetal acceleration, a, is given by this formula, where v is the velocity of the object and r is the object’s distance from the center of the circular path:
a=v^2/r
.

Solve the formula for r.

pls give me the height only!!

Need help with questions 3

White Kittens
Gray Kittens
Male
3
2.
Female
5
5
Find the ratio of white male and gray male kittens to white female and gray female kittens.
O 3: 2:5:5
O 5:7
05:10
03:5
2 3 4
02
5
5
7
8
9
1
https://weskids.schoology.com/common-assessment-delivery/start/4718034729?action=onresume&submissionid=4319101503
Type here to search
SI

7) 4r^2- 12r + 5=0 in quadratic form​

y = 5x - 9
y - 5x = 4
(3,2)
no solution
an infinite number of solutions

The graph of f(x) = x+ was transformed to create the graph of g(x) = f(x) - 9.
Which statement about the graphs is true?
A)
The graph of g is a reflection of the graph of facross the x-axis.
B)
The vertex of the graph of g is 9 units to the right of the vertex of the
graph of f.
C)
The graph of g is a reflection of the graph of facross the y-axis.
D)
The y-intercept of the graph of g is 9 units below the y-intercept of the
graph of f.

Fraction 2 1/3 ÷ 1/4​

A triangle has an area of 36 square meters. What could its base and height be?

b = 12 m, h = 6 m
b = 9 m, h = 4 m
b = 9 m, h = 9 m
b = 8 m, h = 9 m

What is the height of the screen?