What is the value of sec^2(arctan(2))+csc^2(arctan(3)) ?

The value of sec^2(arctan(2))+csc^2(arctan(3)) is 55/9

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sec^2(arctan(2))+csc^2(arctan(3))

sec^{2} (arctan (2)) + csc^{2} (arctan (3))

\frac{55}{9}

Decimal Notation

6.11111 \dots

Solution steps

sec^{2} (arctan (2)) + csc^{2} (arctan (3))

Rewrite using trig identities:

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

Use the following identity:

csc (arctan (x)) = \frac{1 + x ^{2}}{x}

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

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

Simplify

= sec^{2} (arctan (2)) + \frac{10}{9}

Rewrite using trig identities:

sec (arctan (2)) = 5

sec (arctan (2))

Rewrite using trig identities:

sec (arctan (2)) = 1 + 2^{2}

Use the following identity:

sec (arctan (x)) = 1 + x^{2}

= 1 + 2^{2}

= 1 + 2^{2}

Simplify

= 5

= (5)^{2} + \frac{10}{9}

Simplify

(5)^{2} + \frac{10}{9} : \frac{55}{9}

(5)^{2} + \frac{10}{9}

(5)^{2} = 5

(5)^{2}

Apply radical rule:

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

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 5^{\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

= 5

= 5 + \frac{10}{9}

Convert element to fraction:

5 = \frac{5 \cdot 9}{9}

= \frac{5 \cdot 9}{9} + \frac{10}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{5 \cdot 9 + 10}{9}

5 \cdot 9 + 10 = 55

5 \cdot 9 + 10

Multiply the numbers:

5 \cdot 9 = 45

= 45 + 10

Add the numbers:

45 + 10 = 55

= 55

= \frac{55}{9}

= \frac{55}{9}

cos (\frac{10 π}{9})

\frac{1}{2} (π - \frac{1}{2} sin (2 π))

cos (arcsin (\frac{1}{5}) + arctan (\frac{1}{3}))

sin (arcsin (\frac{5}{13}) + arccos (- \frac{3}{5}))

sin (\frac{π}{2}) - 2

What is the value of sec^2(arctan(2))+csc^2(arctan(3)) ?
The value of sec^2(arctan(2))+csc^2(arctan(3)) is 55/9