Prove that,
(1+sin theta + cos theta) ^2 = 2(1+sin theta) (1+cos theta)

Proof below

Step-by-step explanation:

Trigonometric Identities

Before we prove the given identity, we need to recall:

We have to prove that:

Let's expand the square on the left side as indicated in the formula above:

Recall the fundamental trigonometric identity:

Substituting and operating:

Factoring by 2:

Factoring cos θ in the last two terms:

Factoring

Proven

76%

step-by-step explanation:

24% are dogs and the remaining is 76% because 100-24=76

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240