What is the value of sec^2(pi/6)-tan^2(pi/6)-cos^2(pi/6) ?

The value of sec^2(pi/6)-tan^2(pi/6)-cos^2(pi/6) is 1/4

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sec^2(pi/6)-tan^2(pi/6)-cos^2(pi/6)

Solution

sec^{2} (\frac{π}{6}) - tan^{2} (\frac{π}{6}) - cos^{2} (\frac{π}{6})

Solution

\frac{1}{4}

Decimal Notation

0.25

Solution steps

sec^{2} (\frac{π}{6}) - tan^{2} (\frac{π}{6}) - cos^{2} (\frac{π}{6})

Use the following trivial identity:

sec (\frac{π}{6}) = \frac{2 3}{3}

sec (\frac{π}{6})

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= \frac{2 3}{3}

Use the following trivial identity:

tan (\frac{π}{6}) = \frac{3}{3}

tan (\frac{π}{6})

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

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

Simplify

(\frac{2 3}{3})^{2} - (\frac{3}{3})^{2} - (\frac{3}{2})^{2} : \frac{1}{4}

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

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

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

Apply exponent rule:

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

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(23)^{2} = 2^{2} (3)^{2}

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

(3)^{2} : 3

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

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

Cancel the common factor:

3

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

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

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

Apply exponent rule:

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

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

(3)^{2} : 3

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

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

Cancel the common factor:

3

= \frac{1}{3}

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

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

Apply exponent rule:

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

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

(3)^{2} : 3

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

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

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

Combine the fractions

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

Apply rule

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

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

2^{2} - 1 = 3

2^{2} - 1

2^{2} = 4

= 4 - 1

Subtract the numbers:

4 - 1 = 3

= 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

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

2^{2} = 4

= 1 - \frac{3}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{3}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

Popular Examples

20*sin(40)

20 \cdot sin (4 0^{\circ})

4sin(4pi)

4 sin (4 π)

tan^2(61)

tan^{2} (6 1^{\circ})

1^2+ln(1)-sin(1)

1^{2} + ln (1) - sin (1)

sin(180^2)

sin (18 0^{2})

Frequently Asked Questions (FAQ)

What is the value of sec^2(pi/6)-tan^2(pi/6)-cos^2(pi/6) ?
The value of sec^2(pi/6)-tan^2(pi/6)-cos^2(pi/6) is 1/4