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3 years ago

Match the right side of the equation to the left side cot x sec^4 x=cot x+ 2 tan x+tan^3 x

Mathematics

1 answer:

Natalka [10]3 years ago

5 0

Step-by-step explanation:

cot x + 2 tan x + tan³ x

Write in terms of sine and cosine:

(cos x / sin x) + 2 (sin x / cos x) + (sin³ x / cos³ x)

Find the common denominator:

(cos⁴ x / (sin x cos³ x) + 2 (sin² x cos² x / (sin x cos³ x)) + (sin⁴ x / (sin x cos³ x))

Add:

(cos⁴ x + 2 sin² x cos² x + sin⁴ x) / (sin x cos³ x)

Factor:

(sin² x + cos² x)² / (sin x cos³ x)

Pythagorean identity:

1 / (sin x cos³ x)

Multiply top and bottom by cos x:

cos x / (sin x cos⁴ x)

Simplify:

cot x sec⁴ x

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2 years ago

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Use the "rule of 72" to estimate the doubling time (in years) for the interest rate, and then calculate it exactly. (Round your

bonufazy [111]

Answer:

Using the rule of 72, the doubling time is 9.35 years.

The exact answer is that the doubling time is 8.89 years.

Step-by-step explanation:

By the rule of 72, we have that the doubling time D is given by:

$D = \frac{72}{Interest Rate}$

The interest rate is in %.

In our exercise, the interest rate is 7.7%. So, by the rule of 72:

$D = \frac{72}{7.7} = 9.35$ .

Exact answer:

The exact answer is going to be found using the compound interest formula(since the rule of 72 is a simplification of this formula).

The compound interest formula is given by:

$A = P(1 + \frac{r}{n})^{nt}$

Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

So, for this exercise, we have:

We want to find the doubling time, that is, the time in which the amount is double the initial amount, double the principal.

$A = 2P$