What is the value of (1-tan(30)sec(30))/(1+sec(60)cot(60)) ?

The value of (1-tan(30)sec(30))/(1+sec(60)cot(60)) is (2sqrt(3)-3)/3

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(1-tan(30)sec(30))/(1+sec(60)cot(60))

Solution

\frac{1 - tan ( 3 0 ^{\circ} ) sec ( 3 0 ^{\circ} )}{1 + sec ( 6 0 ^{\circ} ) cot ( 6 0 ^{\circ} )}

Solution

\frac{2 3 - 3}{3}

Decimal Notation

0.15470 \dots

Solution steps

\frac{1 - tan ( 3 0 ^{\circ} ) sec ( 3 0 ^{\circ} )}{1 + sec ( 6 0 ^{\circ} ) cot ( 6 0 ^{\circ} )}

Use the following trivial identity:

tan (3 0^{\circ}) = \frac{3}{3}

tan (3 0^{\circ})

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

Use the following trivial identity:

sec (3 0^{\circ}) = \frac{2 3}{3}

sec (3 0^{\circ})

sec (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= \frac{2 3}{3}

Use the following trivial identity:

sec (6 0^{\circ}) = 2

sec (6 0^{\circ})

sec (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 2

Use the following trivial identity:

cot (6 0^{\circ}) = \frac{3}{3}

cot (6 0^{\circ})

cot (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= \frac{3}{3}

= \frac{1 - \frac{3}{3} \cdot \frac{2 3}{3}}{1 + 2 \cdot \frac{3}{3}}

Simplify

\frac{1 - \frac{3}{3} \cdot \frac{2 3}{3}}{1 + 2 \cdot \frac{3}{3}} : \frac{2 3 - 3}{3}

\frac{1 - \frac{3}{3} \cdot \frac{2 3}{3}}{1 + 2 \cdot \frac{3}{3}}

Multiply

2 \cdot \frac{3}{3} : \frac{2 3}{3}

2 \cdot \frac{3}{3}

Multiply fractions:

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

= \frac{3 \cdot 2}{3}

= \frac{1 - \frac{3}{3} \cdot \frac{2 3}{3}}{1 + \frac{2 3}{3}}

\frac{3}{3} \cdot \frac{2 3}{3} = \frac{2}{3}

\frac{3}{3} \cdot \frac{2 3}{3}

Multiply fractions:

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

= \frac{3 \cdot 2 3}{3 \cdot 3}

3 \cdot 23 = 6

3 \cdot 23

Apply radical rule:

a a = a

33 = 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{6}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{6}{9}

Cancel the common factor:

3

= \frac{2}{3}

= \frac{1 - \frac{2}{3}}{1 + \frac{2 3}{3}}

Join

1 + \frac{3 \cdot 2}{3} : \frac{3 + 2}{3}

1 + \frac{3 \cdot 2}{3}

Convert element to fraction:

1 = \frac{1 \cdot 3}{3}

= \frac{1 \cdot 3}{3} + \frac{3 \cdot 2}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 3 + 3 \cdot 2}{3}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3 + 2 3}{3}

Factor

3 + 32 : 3 (3 + 2)

3 + 3 \cdot 2

3 = 33

= 33 + 3 \cdot 2

Factor out common term

3

= 3 (3 + 2)

= \frac{3 ( 3 + 2 )}{3}

Cancel

\frac{3 ( 3 + 2 )}{3} : \frac{3 + 2}{3}

\frac{3 ( 3 + 2 )}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}} ( 2 + 3 )}{3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{3 ^{\frac{1}{2}}}{3 ^{1}} = \frac{1}{3 ^{1 - \frac{1}{2}}}

= \frac{3 + 2}{3 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{3 + 2}{3 ^{\frac{1}{2}}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{3 + 2}{3}

= \frac{3 + 2}{3}

= \frac{1 - \frac{2}{3}}{\frac{3 + 2}{3}}

Join

1 - \frac{2}{3} : \frac{1}{3}

1 - \frac{2}{3}

Convert element to fraction:

1 = \frac{1 \cdot 3}{3}

= \frac{1 \cdot 3}{3} - \frac{2}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 3 - 2}{3}

1 \cdot 3 - 2 = 1

1 \cdot 3 - 2

Multiply the numbers:

1 \cdot 3 = 3

= 3 - 2

Subtract the numbers:

3 - 2 = 1

= 1

= \frac{1}{3}

= \frac{\frac{1}{3}}{\frac{3 + 2}{3}}

Divide fractions:

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

= \frac{1 \cdot 3}{3 ( 3 + 2 )}

Refine

= \frac{3}{3 ( 3 + 2 )}

Rationalize

\frac{3}{3 ( 2 + 3 )} : \frac{2 3 - 3}{3}

\frac{3}{3 ( 2 + 3 )}

Multiply by the conjugate

\frac{3 - 2}{3 - 2}

= \frac{3 ( 3 - 2 )}{3 ( 3 + 2 ) ( 3 - 2 )}

3 (3 - 2) = 3 - 23

3 (3 - 2)

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 3, c = 2

= 33 - 3 \cdot 2

= 33 - 23

Apply radical rule:

a a = a

33 = 3

= 3 - 23

3 (3 + 2) (3 - 2) = - 3

3 (3 + 2) (3 - 2)

Expand

(3 + 2) (3 - 2) : - 1

(3 + 2) (3 - 2)

Apply Difference of Two Squares Formula:

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

a = 3, b = 2

= (3)^{2} - 2^{2}

Simplify

(3)^{2} - 2^{2} : - 1

(3)^{2} - 2^{2}

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

2^{2} = 4

2^{2}

2^{2} = 4

= 4

= 3 - 4

Subtract the numbers:

3 - 4 = - 1

= - 1

= - 1

= 3 (- 1)

Expand

3 (- 1) : - 3

3 (- 1)

Distribute parentheses

= 3 (- 1)

Apply minus-plus rules

+ (- a) = - a

= - 3 \cdot 1

Multiply the numbers:

3 \cdot 1 = 3

= - 3

= - 3

= \frac{3 - 2 3}{- 3}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

3 - 23 = - (23 - 3)

= \frac{2 3 - 3}{3}

= \frac{2 3 - 3}{3}

= \frac{2 3 - 3}{3}

Popular Examples

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sin (\frac{25}{30})

Frequently Asked Questions (FAQ)

What is the value of (1-tan(30)sec(30))/(1+sec(60)cot(60)) ?
The value of (1-tan(30)sec(30))/(1+sec(60)cot(60)) is (2sqrt(3)-3)/3