Is cos(b)(1+tan(b))^2=sec(b)+2sin(b) ?

The answer to whether cos(b)(1+tan(b))^2=sec(b)+2sin(b) is True

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prove cos(b)(1+tan(b))^2=sec(b)+2sin(b)

Solution

prove

cos (b) (1 + tan (b))^{2} = sec (b) + 2 sin (b)

Solution

True

Solution steps

cos (b) (1 + tan (b))^{2} = sec (b) + 2 sin (b)

Manipulating left side

cos (b) (1 + tan (b))^{2}

Express with sin, cos

(1 + tan (b))^{2} cos (b)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

Simplify

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

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

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

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

Join

1 + \frac{sin ( b )}{cos ( b )} : \frac{cos ( b ) + sin ( b )}{cos ( b )}

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

Convert element to fraction:

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

= \frac{1 \cdot cos ( b )}{cos ( b )} + \frac{sin ( b )}{cos ( b )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot cos ( b ) + sin ( b )}{cos ( b )}

Multiply:

1 \cdot cos (b) = cos (b)

= \frac{cos ( b ) + sin ( b )}{cos ( b )}

= (\frac{cos ( b ) + sin ( b )}{cos ( b )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( cos ( b ) + sin ( b ) ) ^{2}}{cos ^{2} ( b )}

= \frac{( cos ( b ) + sin ( b ) ) ^{2}}{cos ^{2} ( b )} cos (b)

Multiply fractions:

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

= \frac{( cos ( b ) + sin ( b ) ) ^{2} cos ( b )}{cos ^{2} ( b )}

Cancel the common factor:

cos (b)

= \frac{( cos ( b ) + sin ( b ) ) ^{2}}{cos ( b )}

= \frac{( cos ( b ) + sin ( b ) ) ^{2}}{cos ( b )}

= \frac{( cos ( b ) + sin ( b ) ) ^{2}}{cos ( b )}

Expand

(cos (b) + sin (b))^{2} : cos^{2} (b) + 2 cos (b) sin (b) + sin^{2} (b)

(cos (b) + sin (b))^{2}

Apply Perfect Square Formula:

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

a = cos (b), b = sin (b)

= cos^{2} (b) + 2 cos (b) sin (b) + sin^{2} (b)

= \frac{cos ^{2} ( b ) + sin ^{2} ( b ) + 2 cos ( b ) sin ( b )}{cos ( b )}

Rewrite using trig identities

\frac{cos ^{2} ( b ) + sin ^{2} ( b ) + 2 cos ( b ) sin ( b )}{cos ( b )}

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= \frac{1 + 2 cos ( b ) sin ( b )}{cos ( b )}

= \frac{1 + 2 cos ( b ) sin ( b )}{cos ( b )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1 + 2 \cdot \frac{1}{s e c ( b )} sin ( b )}{\frac{1}{s e c ( b )}}

Simplify

\frac{1 + 2 \cdot \frac{1}{s e c ( b )} sin ( b )}{\frac{1}{s e c ( b )}}

Apply the fraction rule:

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

= \frac{( 1 + 2 \cdot \frac{1}{s e c ( b )} sin ( b ) ) sec ( b )}{1}

2 \cdot \frac{1}{sec ( b )} sin (b) = \frac{2 sin ( b )}{sec ( b )}

2 \cdot \frac{1}{sec ( b )} sin (b)

Multiply fractions:

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

= \frac{1 \cdot 2 sin ( b )}{sec ( b )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 sin ( b )}{sec ( b )}

= \frac{sec ( b ) ( \frac{2 s i n ( b )}{s e c ( b )} + 1 )}{1}

Join

1 + \frac{2 sin ( b )}{sec ( b )} : \frac{sec ( b ) + 2 sin ( b )}{sec ( b )}

1 + \frac{2 sin ( b )}{sec ( b )}

Convert element to fraction:

1 = \frac{1 sec ( b )}{sec ( b )}

= \frac{1 \cdot sec ( b )}{sec ( b )} + \frac{2 sin ( b )}{sec ( b )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot sec ( b ) + 2 sin ( b )}{sec ( b )}

Multiply:

1 \cdot sec (b) = sec (b)

= \frac{sec ( b ) + 2 sin ( b )}{sec ( b )}

= \frac{\frac{s e c ( b ) + 2 s i n ( b )}{s e c ( b )} sec ( b )}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{sec ( b ) + 2 sin ( b )}{sec ( b )} sec (b)

Multiply fractions:

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

= \frac{( sec ( b ) + 2 sin ( b ) ) sec ( b )}{sec ( b )}

Cancel the common factor:

sec (b)

= sec (b) + 2 sin (b)

sec (b) + 2 sin (b)

sec (b) + 2 sin (b)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cos(b)(1+tan(b))^2=sec(b)+2sin(b) ?
The answer to whether cos(b)(1+tan(b))^2=sec(b)+2sin(b) is True