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Sloan [31]
3 years ago
9

Prove the identity cos x/1 - sin x = sec x + tan x

Mathematics
1 answer:
kompoz [17]3 years ago
7 0

Answer:

Proved

Step-by-step explanation:

Given

Prove that

\frac{cos x}{1 - sin x} = sec x + tan x

\frac{cos x}{1 - sin x}

Multiply the numerator and denominator by 1 + sinx

\frac{cos x}{1 - sin x} * \frac{1 + sin x}{1 + sin x}

Combine both fractions to form 1

\frac{cos x (1 + sin x)}{(1 - sin x)(1 + sin x)}

Expand the denominator using difference of two squares;

i.e.\ (a - b)(a + b) = a^2 - b^2

The expression becomes

\frac{cos x (1 + sin x)}{(1^2 - sin^2 x)}

\frac{cos x (1 + sin x)}{(1 - sin^2 x)}

From trigonometry; 1 - sin^2x = cos^2x

The expression becomes

\frac{cos x (1 + sin x)}{(cos^2 x)}

Divide the numerator and the denominator by cos x

\frac{(1 + sin x)}{(cos x)}

Split fraction

\frac{1}{cos x} + \frac{sin x}{cos x}

From trigonometry; \frac{1}{cos x} = sec x \ and\ \frac{sin\ x}{cos\ x} = tan\ x

So;

\frac{1}{cos x} + \frac{sin x}{cos x} = sec x + tan x

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Lady bird [3.3K]

Answer:

The amount Tobie invests in a bank, P = $25

The annual compound interest rate, i = 4%

a. The principal is amount invested in the bank, P = $25

The annual interest rate, i = 4%

b. The function that represents Tobie's account balance, <em>A</em>, after <em>t</em> years is given as follows;

A(t) = P \cdot (1 + i)^t

Where;

A(t) = The amount in the account after <em>t</em> years

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i = The annual compound interest rate

t = The number of years

c. The values to place in the table are found as follows;

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At t = 10, A(10) = 25 \times(1 + 0.04)^{10} = 37

The given table is presented as follows;

\begin{array}{ccc}t&&A(t)\\0&&25\\10&&37\end{array}

Step-by-step explanation:

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A coin collection is made up of 34 coins comprised of nickels and dimes. The total value of the collection is $1.90. How many di
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Answer:

x = nickels

y= dimes

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