45 times the numerator plus the denominator is most likely going to equal the same amount as if it were doubled.
Answer: see proof below
<u>Step-by-step explanation:</u>
Given: A + B + C = π and cos A = cos B · cos C
scratchwork:
A + B + C = π
A = π - (B + C)
cos A = cos [π - (B + C)] Apply cos
= - cos (B + C) Simplify
= -(cos B · cos C - sin B · sin C) Sum Identity
= sin B · sin C - cos B · cos C Simplify
cos B · cos C = sin B · sin C - cos B · cos C Substitution
2cos B · cos C = sin B · sin C Addition
Division
2 = tan B · tan C

<u>Proof LHS → RHS</u>
Given: A + B + C = π
Subtraction: A = π - (B + C)
Apply tan: tan A = tan(π - (B + C))
Simplify: = - tan (B + C)

Substitution: = -(tan B + tan C)/(1 - 2)
Simplify: = -(tan B + tan C)/-1
= tan B + tan C
LHS = RHS: tan B + tan C = tan B + tan C 
Answer: 
Step-by-step explanation:
Since, The total number of student = 300
Out of which,
The number of students who are only in Maths = 120
And, The number of students who are only in Science = 50
While, the students who are not from any subject = 100
Hence, the number of student who are from both maths and science = Total student - Maths student (only) - science student (only) - None
= 300 - 120 - 50 - 100
= 30
That is, there are 30 students who are both from science and maths,
Thus, the probability of selecting one student who is both from maths and science = 30/300 = 1/10
Formula for growth:
y = initial • (percent growth + 1)^time/frequency of growth
So, since initial population is 382
Percent growth is 0.007
The time is 30 years
And it goes it by .07% every 1 year,
y = 382 • (1.007)^30/1
y= 470.9203732 people
Since we can’t have 0.9 of a person, the population after 30 years will be 470 people.