What is the value of tan^2(30)-sec^2(30) ?

The value of tan^2(30)-sec^2(30) is -1

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tan^2(30)-sec^2(30)

Solution

tan^{2} (3 0^{\circ}) - sec^{2} (3 0^{\circ})

Solution

- 1

Solution steps

tan^{2} (3 0^{\circ}) - sec^{2} (3 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}

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

Simplify

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

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

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

Apply rule

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

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

1 - 2^{2} = - 3

1 - 2^{2}

2^{2} = 4

= 1 - 4

Subtract the numbers:

1 - 4 = - 3

= - 3

= \frac{- 3}{3}

Apply the fraction rule:

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

= - \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= - 1

= - 1

Popular Examples

tan(2arccos(1/4))

tan (2 arccos (\frac{1}{4}))

(sin(30)+cos(30))/(tan(30))

\frac{sin ( 3 0 ^{\circ} ) + cos ( 3 0 ^{\circ} )}{tan ( 3 0 ^{\circ} )}

4cos(30)+6sin(45)

4 cos (3 0^{\circ}) + 6 sin (4 5^{\circ})

tan^2((5pi)/4)

tan^{2} (\frac{5 π}{4})

5000cos(30)

5000 cos (3 0^{\circ})

Frequently Asked Questions (FAQ)

What is the value of tan^2(30)-sec^2(30) ?
The value of tan^2(30)-sec^2(30) is -1