What is the value of (1+cot^2(30))/(1+tan^2(30)) ?

The value of (1+cot^2(30))/(1+tan^2(30)) is 3

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(1+cot^2(30))/(1+tan^2(30))

Solution

\frac{1 + cot ^{2} ( 3 0 ^{\circ} )}{1 + tan ^{2} ( 3 0 ^{\circ} )}

Solution

3

Solution steps

\frac{1 + cot ^{2} ( 3 0 ^{\circ} )}{1 + tan ^{2} ( 3 0 ^{\circ} )}

Use the following trivial identity:

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

cot (3 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

= 3

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}

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

Simplify

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

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

(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

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

Add the numbers:

1 + 3 = 4

= \frac{4}{1 + \frac{1}{3}}

Join

1 + \frac{1}{3} : \frac{4}{3}

1 + \frac{1}{3}

Convert element to fraction:

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

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

1 \cdot 3 + 1 = 4

1 \cdot 3 + 1

Multiply the numbers:

1 \cdot 3 = 3

= 3 + 1

Add the numbers:

3 + 1 = 4

= 4

= \frac{4}{3}

= \frac{4}{\frac{4}{3}}

Apply the fraction rule:

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

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

Multiply the numbers:

4 \cdot 3 = 12

= \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3

= 3

Popular Examples

arctan(8/(-4))

arctan (\frac{8}{- 4})

1+tan^2(30)-csc^2(45)

1 + tan^{2} (3 0^{\circ}) - csc^{2} (4 5^{\circ})

cos(2*2pi)

cos (2 \cdot 2 π)

-tan((sqrt(3))/3)

- tan (\frac{3}{3})

cos(-(7pi)/8)

cos (- \frac{7 π}{8})

Frequently Asked Questions (FAQ)

What is the value of (1+cot^2(30))/(1+tan^2(30)) ?
The value of (1+cot^2(30))/(1+tan^2(30)) is 3