Mathematics: Someone please help me to prove this.

Someone please help me to prove this. - id7409580, Theresab2021

AniyahM17

01.02.2022

see proof below

Step-by-step explanation:

Use the following Product to Sum Identities:

2 sin A · cos B = sin (A + B) + sin (A - B)

2 cos A · cos B = sin (A + B) + sin (A - B)

Given: cos A + cos B = 1/2 and sin A + sin B = 1/4

Proof LHS → RHS

$\text{LHS:}\qquad \qquad \qquad \tan\dfrac{A+B}{2}$

$\text{Expand:}\qquad \qquad \dfrac{\sin\frac{(A+B)}{2}}{\cos\frac{(A+B)}{2}}$

$\text{Multiplication:}\qquad \quad \dfrac{\sin\frac{(A+B)}{2}}{\cos\frac{(A+B)}{2}}\bigg(\dfrac{2\cos\frac{A-B}{2}}{2\cos \frac{A-B}{2}}\bigg)$

$\text{Simplify:}\qquad \qquad \quad \dfrac{2\sin \frac{A+B}{2}\cdot \cos \frac{A-B}{2}}{2\cos \frac{A+B}{2}\cdot \cos \frac{A-B}{2}}$

$\text{Product to Sum:}\qquad \dfrac{\sin A+\sin B}{\cos A+\cos B}$

$\text{Given:}\qquad \qquad \qquad \quad \dfrac{\frac{1}{4}}{\frac{1}{2}}$

$\text{Simplify:}\qquad \qquad \qquad \dfrac{1}{2}$

LHS = RHS: 1/2 = 1/2 $\checkmark$

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giusto1073

21.10.2022

pH = f(x) = -log₁₀x

1. Graphs

I used Excel to calculate the pH values and draw the graphs (see the Figure).

f(x) and f(x) +1 are plotted against the left-hand axis, while f(x+ 1) is plotted against the expanded right-hand axis.

The points at which pH = 0 and pH = 1 are indicated by the large red dots.

2. x = 0.5

When x = 0.5, pH ≈0.30. The point is indicated by the red diamond.

3. Transformations

(a) ƒ(x) = -log(x) + 1

This function has no y-intercept, because log(0) is undefined.

(b) ƒ(x +1) = -log(x + 1)

f(0) = -log(0 + 1) = -log(1) = 0

This function has a y-intercept at (0,0).

hope this helps please mark me brainliest!

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