What is the value of 4sec^2(45)tan^2(45)+2sec^4(45) ?

The value of 4sec^2(45)tan^2(45)+2sec^4(45) is 16

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4sec^2(45)tan^2(45)+2sec^4(45)

4 sec^{2} (4 5^{\circ}) tan^{2} (4 5^{\circ}) + 2 sec^{4} (4 5^{\circ})

16

Solution steps

4 sec^{2} (4 5^{\circ}) tan^{2} (4 5^{\circ}) + 2 sec^{4} (4 5^{\circ})

Use the following trivial identity:

sec (4 5^{\circ}) = 2

sec (4 5^{\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:

tan (4 5^{\circ}) = 1

tan (4 5^{\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}

= 1

= 4 (2)^{2} \cdot 1^{2} + 2 (2)^{4}

Simplify

4 (2)^{2} \cdot 1^{2} + 2 (2)^{4} : 16

4 (2)^{2} \cdot 1^{2} + 2 (2)^{4}

Apply rule

1^{a} = 1

1^{2} = 1

= 4 \cdot 1 \cdot (2)^{2} + 2 (2)^{4}

4 (2)^{2} \cdot 1 = 8

4 (2)^{2} \cdot 1

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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

= 2

= 4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 8

2 (2)^{4} = 2^{3}

2 (2)^{4}

(2)^{4} = 2^{2}

(2)^{4}

Apply radical rule:

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

= (2^{\frac{1}{2}})^{4}

Apply exponent rule:

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

= 2^{\frac{1}{2} \cdot 4}

\frac{1}{2} \cdot 4 = 2

\frac{1}{2} \cdot 4

Multiply fractions:

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

= \frac{1 \cdot 4}{2}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 2^{2}

= 2^{2} \cdot 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2^{2} = 2^{1 + 2}

= 2^{1 + 2}

Add the numbers:

1 + 2 = 3

= 2^{3}

= 8 + 2^{3}

2^{3} = 8

= 8 + 8

Add the numbers:

8 + 8 = 16

= 16

= 16

arctan (\frac{13}{11})

cos (- \frac{5}{6})

arccos (1.24)

sec (\frac{1}{4})

3 cot (\frac{5 π}{6})

What is the value of 4sec^2(45)tan^2(45)+2sec^4(45) ?
The value of 4sec^2(45)tan^2(45)+2sec^4(45) is 16