${\mathrm{tan}}^{2}\left(\alpha \right)-{\mathrm{sin}}^{2}\left(\alpha \right)={\mathrm{tan}}^{2}\left(\alpha \right){\mathrm{sin}}^{2}\left(\alpha \right)$ Start on the left side.
${\mathrm{tan}}^{2}\left(\alpha \right)-{\mathrm{sin}}^{2}\left(\alpha \right)$
Convert to sines and cosines.
Write $\mathrm{tan}\left(\alpha \right)$ in sines and cosines using the quotient identity.
${\left(\frac{\mathrm{sin}\left(\alpha \right)}{\mathrm{cos}\left(\alpha \right)}\right)}^{2}-{\mathrm{sin}}^{2}\left(\alpha \right)$
Apply the product rule to $\frac{\mathrm{sin}\left(\alpha \right)}{\mathrm{cos}\left(\alpha \right)}$.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}-{\mathrm{sin}}^{2}\left(\alpha \right)$
Write $-{\mathrm{sin}}^{2}\left(\alpha \right)$ as a fraction with denominator $1$.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}+\frac{-{\mathrm{sin}}^{2}\left(\alpha \right)}{1}$
Add fractions.
To write $\frac{-{\mathrm{sin}}^{2}\left(\alpha \right)}{1}$ as a fraction with a common denominator, multiply by $\frac{{\mathrm{cos}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}+\frac{-{\mathrm{sin}}^{2}\left(\alpha \right)}{1}\cdot \frac{{\mathrm{cos}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$
Multiply $\frac{-{\mathrm{sin}}^{2}\left(\alpha \right)}{1}$ by $\frac{{\mathrm{cos}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}+\frac{-{\mathrm{sin}}^{2}\left(\alpha \right){\mathrm{cos}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$
Combine the numerators over the common denominator.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)-{\mathrm{sin}}^{2}\left(\alpha \right){\mathrm{cos}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$
Apply Pythagorean identity.
Multiply by $1$.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)\cdot 1-{\mathrm{sin}}^{2}\left(\alpha \right){\mathrm{cos}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$
Factor ${\mathrm{sin}}^{2}\left(\alpha \right)$ out of $-{\mathrm{sin}}^{2}\left(\alpha \right){\mathrm{cos}}^{2}\left(\alpha \right)$.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)\cdot 1+{\mathrm{sin}}^{2}\left(\alpha \right)(-{\mathrm{cos}}^{2}\left(\alpha \right))}{{\mathrm{cos}}^{2}\left(\alpha \right)}$
Factor ${\mathrm{sin}}^{2}\left(\alpha \right)$ out of ${\mathrm{sin}}^{2}\left(\alpha \right)\cdot 1+{\mathrm{sin}}^{2}\left(\alpha \right)(-{\mathrm{cos}}^{2}\left(\alpha \right))$.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)\cdot (1-{\mathrm{cos}}^{2}\left(\alpha \right))}{{\mathrm{cos}}^{2}\left(\alpha \right)}$
Apply pythagorean identity.
$\frac{{\mathrm{sin}}^{2}\left(\alpha \right)\cdot {\mathrm{sin}}^{2}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$
Multiply$\mathrm{sin}\left(\alpha \right)}^{2$ by $\mathrm{sin}\left(\alpha \right)}^{2$ by adding the exponents.
$\frac{{\mathrm{sin}}^{4}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$
Rewrite $\frac{{\mathrm{sin}}^{4}\left(\alpha \right)}{{\mathrm{cos}}^{2}\left(\alpha \right)}$ as ${\mathrm{tan}}^{2}\left(\alpha \right){\mathrm{sin}}^{2}\left(\alpha \right)$.
${\mathrm{tan}}^{2}\left(\alpha \right){\mathrm{sin}}^{2}\left(\alpha \right)$
Because the two sides have been shown to be equivalent, the equation is an identity.
${\mathrm{tan}}^{2}\left(\alpha \right)-{\mathrm{sin}}^{2}\left(\alpha \right)={\mathrm{tan}}^{2}\left(\alpha \right){\mathrm{sin}}^{2}\left(\alpha \right)$ is an identity
The combined area of a square and a rectangle is 225 square yards. the length of the rectangle is 8 times the width of the rectangle and the length of a side of the square is the same as the width of the rectangle. find the dimensions of the square and the rectangle.
Help how do i find the poralitlbity that 2 kings are dealt in a 5 card hand from a standard 52 card deck??
1. Random sample of size 4 are drawn from the finite population which consists of the numbers 2,3,7,8,and 10. a. What is the population mean, population variance and population standard deviation of the given data? b. What is the sampling distribution of the sample means for a sample of size 2 which can be drawn without replacement from the given population? c. What is the mean, variance and standard deviation of the samp
Finding a constant such that the following integral inequality holds:
Constant: $c>0$ such that for all $C}^{1$ function in $(0,1)$
$cu{\left(0\right)}^{2}\le {\int}_{0}^{1}{u}^{\prime 2}+{u}^{2}dt$
PROBLEM: An archaeologist has conducted an archaeological surface survey in a region of western New Zealand consisting of a large and fertile valley with a major river flowing down to the coast. The archaeologist has hypothesized that the extremely complex Maori chiefdoms that the Europeans encountered upon contact in the 17th century were actually relatively recent developments in response to population pressure in the latest prehistoric phase. Further, she has argued that what was once a coastally-focused chiefdom oriented towards a maritime economy, became increasingly inland-focused over time, with a subsistence emphasis on root crop agriculture rather than marine resource exploitation in the latest phase.
Sites appear on the surface as large quantities of subsistence debris and lithic artifacts (they had no ceramics), without standing architecture except for readily visible stone ceremonial platforms. The archaeologist has identified two major cultural phases in the region: (1) PHASE I (dated to about A.D. 1000-1300) and (2) PHASE II (dated to about A.D. 1300-1500) from 3 different sites. The archaeologist would like to examine the variable of “population size” in two ways: the absolute size of the site and the number of stone ceremonial platforms. The archaeologist has also reasoned that she can get into the issue of relative reliance on agriculture vs. maritime subsistence by looking at both the distance of sites from the coast and the relative density of fish bones, shell and other marine resources in the surface remains at the site before and after a long period of time. The data collected are as follows:
Site | Period | Size (in ha.) | # Ceremonial Structures | Distance to Coast (in km) | % Marine Resources(before) | % Marine Resources(after) |
1 | 1 | 3.40 | 5.00 | 3.20 | 61.00 | 58.00 |
2 | 1 | 9.80 | 7.00 | 1.20 | 56.00 | 55.00 |
3 | 1 | 4.20 | 6.00 | 3.30 | 54.00 | 52.00 |
1 | 1 | 1.20 | 2.00 | 7.30 | 31.00 | 29.00 |
2 | 1 | 3.30 | 6.00 | 4.40 | 61.00 | 60.00 |
3 | 1 | 2.50 | 4.00 | 5.30 | 45.00 | 45.00 |
1 | 1 | 5.40 | 5.00 | 2.10 | 58.00 | 58.00 |
2 | 1 | 1.60 | 2.00 | 6.80 | 46.00 | 45.00 |
3 | 1 | 2.80 | 5.00 | 5.80 | 47.00 | 46.00 |
1 | 1 | 4.70 | 6.00 | 3.40 | 51.00 | 50.00 |
2 | 1 | 3.60 | 4.00 | 4.40 | 62.00 | 61.00 |
3 | 1 | 9.70 | 3.00 | 2.40 | 53.00 | 53.00 |
1 | 1 | 2.20 | 2.00 | 6.70 | 32.00 | 30.00 |
2 | 1 | 2.80 | 3.00 | 5.20 | 61.00 | 60.00 |
3 | 1 | 2.90 | 4.00 | 4.20 | 67.00 | 66.00 |
1 | 2 | 5.40 | 5.00 | 1.30 | 56.00 | 55.00 |
2 | 2 | 3.30 | 2.00 | 9.30 | 16.00 | 14.00 |
3 | 2 | 9.40 | 6.00 | 7.50 | 32.00 | 30.00 |
1 | 2 | 8.20 | 4.00 | 5.80 | 45.00 | 43.00 |
2 | 2 | 13.40 | 8.00 | 7.80 | 34.00 | 30.00 |
3 | 2 | 2.20 | 2.00 | 8.50 | 26.00 | 25.00 |
1 | 2 | 6.50 | 5.00 | 5.60 | 42.00 | 42.00 |
2 | 2 | 7.30 | 5.00 | 6.30 | 41.00 | 40.00 |
3 | 2 | 4.30 | 4.00 | 8.90 | 24.00 | 23.00 |
1 | 2 | 4.10 | 3.00 | 9.10 | 12.00 | 11.00 |
2 | 2 | 7.10 | 4.00 | 7.10 | 48.00 | 46.00 |
3 | 2 | 9.50 | 6.00 | 7.50 | 36.00 | 35.00 |
1 | 2 | 3.50 | 4.00 | 3.20 | 57.00 | 56.00 |
2 | 2 | 10.30 | 6.00 | 6.20 | 34.00 | 33.00 |
3 | 2 | 2.50 | 3.00 | 4.50 | 46.00 | 46.00 |
1 | 2 | 5.60 | 4.00 | 5.30 | 49.00 | 49.00 |
2 | 2 | 9.10 | 5.00 | 6.50 | 35.00 | 35.00 |
3 | 2 | 9.90 | 7.00 | 7.20 | 22.00 | 22.00 |
1 | 2 | 3.20 | 3.00 | 4.30 | 54.00 | 53.00 |
2 | 2 | 9.20 | 7.00 | 7.10 | 35.00 | 30.00 |
3 | 2 | 5.30 | 5.00 | 8.30 | 32.00 | 32.00 |
1 | 2 | 4.90 | 5.00 | 9.50 | 21.00 | 20.00 |
2 | 2 | 5.10 | 5.00 | 10.20 | 12.00 | 15.00 |
3 | 2 | 6.10 | 6.00 | 8.20 | 23.00 | 23.00 |
1 | 2 | 6.80 | 6.00 | 6.70 | 33.00 | 30.00 |
a. Using the variable “ceremonial structures’, plot a histogram, run the descriptives and interpret the results.
b. Construct a frequency distribution table for the sites and periods then interpret.
c. Construct a 95% Confidence Interval for the variable “distance to coast (in km)” then interpret.
d. Is there a reason to believe that the sample average site size (in ha) is the same as the population site size of 6 ha? Run the appropriate statistical tool and interpret.
e. The archaeologist suspects that the percentage of marine resources decreased after a long period of time. Is there a reason to believe in the archaeologist’s claim?
f. Is there a significant difference in the site size between the two periods?
g. Is there a significant difference in the ceremonial structures between the three different sites?
What method is required to find out "for what values of k is $4{x}^{2}+kx+\frac{1}{4}$ a perfect square?"