Is ((cos^2(x)+4cos(x)+4))/((cos(x)+2))=((2sec(x)+1))/(sec(x)) ?

The answer to whether ((cos^2(x)+4cos(x)+4))/((cos(x)+2))=((2sec(x)+1))/(sec(x)) is True

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Popular Trigonometry >

prove ((cos^2(x)+4cos(x)+4))/((cos(x)+2))=((2sec(x)+1))/(sec(x))

Solution

prove

\frac{( cos ^{2} ( x ) + 4 cos ( x ) + 4 )}{( cos ( x ) + 2 )} = \frac{( 2 sec ( x ) + 1 )}{sec ( x )}

Solution

True

Solution steps

\frac{cos ^{2} ( x ) + 4 cos ( x ) + 4}{cos ( x ) + 2} = \frac{2 sec ( x ) + 1}{sec ( x )}

Manipulating left side

\frac{cos ^{2} ( x ) + 4 cos ( x ) + 4}{cos ( x ) + 2}

Simplify

\frac{4 + cos ^{2} ( x ) + 4 cos ( x )}{2 + cos ( x )} : cos (x) + 2

\frac{4 + cos ^{2} ( x ) + 4 cos ( x )}{2 + cos ( x )}

Factor

4 + cos^{2} (x) + 4 cos (x) : (cos (x) + 2)^{2}

4 + cos^{2} (x) + 4 cos (x)

Write in the standard form

a x^{2} + b x + c

= cos^{2} (x) + 4 cos (x) + 4

Rewrite

cos^{2} (x) + 4 cos (x) + 4

cos^{2} (x) + 2 cos (x) \cdot 2 + 2^{2}

cos^{2} (x) + 4 cos (x) + 4

Rewrite

4

2^{2}

= cos^{2} (x) + 2^{2} cos (x) + 2^{2}

Rewrite

2^{2} cos (x)

2 cos (x) \cdot 2

= cos^{2} (x) + 2 cos (x) \cdot 2 + 2^{2}

= cos^{2} (x) + 2 cos (x) \cdot 2 + 2^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = cos (x), b = 2

= (cos (x) + 2)^{2}

= \frac{( cos ( x ) + 2 ) ^{2}}{2 + cos ( x )}

Cancel the common factor:

cos (x) + 2

= cos (x) + 2

= 2 + cos (x)

Manipulating right side

\frac{2 sec ( x ) + 1}{sec ( x )}

Express with sin, cos

\frac{1 + 2 sec ( x )}{sec ( x )}

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1 + 2 \cdot \frac{1}{c o s ( x )}}{\frac{1}{c o s ( x )}}

Simplify

\frac{1 + 2 \cdot \frac{1}{c o s ( x )}}{\frac{1}{c o s ( x )}} : cos (x) + 2

\frac{1 + 2 \cdot \frac{1}{c o s ( x )}}{\frac{1}{c o s ( x )}}

Apply the fraction rule:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

= \frac{( 1 + 2 \cdot \frac{1}{c o s ( x )} ) cos ( x )}{1}

2 \cdot \frac{1}{cos ( x )} = \frac{2}{cos ( x )}

2 \cdot \frac{1}{cos ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{cos ( x )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{cos ( x )}

= \frac{cos ( x ) ( \frac{2}{c o s ( x )} + 1 )}{1}

Join

1 + \frac{2}{cos ( x )} : \frac{cos ( x ) + 2}{cos ( x )}

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

Convert element to fraction:

1 = \frac{1 cos ( x )}{cos ( x )}

= \frac{1 \cdot cos ( x )}{cos ( x )} + \frac{2}{cos ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot cos ( x ) + 2}{cos ( x )}

Multiply:

1 \cdot cos (x) = cos (x)

= \frac{cos ( x ) + 2}{cos ( x )}

= \frac{\frac{c o s ( x ) + 2}{c o s ( x )} cos ( x )}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{cos ( x ) + 2}{cos ( x )} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( cos ( x ) + 2 ) cos ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= cos (x) + 2

= cos (x) + 2

= 2 + cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is ((cos^2(x)+4cos(x)+4))/((cos(x)+2))=((2sec(x)+1))/(sec(x)) ?
The answer to whether ((cos^2(x)+4cos(x)+4))/((cos(x)+2))=((2sec(x)+1))/(sec(x)) is True