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c= 1/(sec(x)+tan(x))+1/(sec(x)-tan(x))

Solution

c = \frac{1}{sec ( x ) + tan ( x )} + \frac{1}{sec ( x ) - tan ( x )}

x = arcsec (\frac{c}{2}) + 2 πn, x = - arcsec (\frac{c}{2}) + 2 πn

Solution steps

c = \frac{1}{sec ( x ) + tan ( x )} + \frac{1}{sec ( x ) - tan ( x )}

Switch sides

\frac{1}{sec ( x ) + tan ( x )} + \frac{1}{sec ( x ) - tan ( x )} = c

Subtract

c

from both sides

- c + \frac{2 sec ( x )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )} = 0

Simplify

- c + \frac{2 sec ( x )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )} : \frac{- c ( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) ) + 2 sec ( x )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )}

- c + \frac{2 sec ( x )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )}

Convert element to fraction:

c = \frac{c ( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )}

= - \frac{c ( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )} + \frac{2 sec ( x )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )}

Since the denominators are equal, combine the fractions:

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

= \frac{- c ( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) ) + 2 sec ( x )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )}

\frac{- c ( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) ) + 2 sec ( x )}{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- c (sec (x) + tan (x)) (sec (x) - tan (x)) + 2 sec (x) = 0

Rewrite using trig identities

- c (sec (x) + tan (x)) (sec (x) - tan (x)) + 2 sec (x)

(sec (x) + tan (x)) (sec (x) - tan (x)) = 1

(sec (x) + tan (x)) (sec (x) - tan (x))

Expand

(sec (x) + tan (x)) (sec (x) - tan (x)) : sec^{2} (x) - tan^{2} (x)

(sec (x) + tan (x)) (sec (x) - tan (x))

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = sec (x), b = tan (x)

= sec^{2} (x) - tan^{2} (x)

= sec^{2} (x) - tan^{2} (x)

Use the Pythagorean identity:

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

sec^{2} (x) - tan^{2} (x) = 1

= 1

= 2 sec (x) - c \cdot 1

2 sec (x) - c \cdot 1 = 0

Multiply:

c \cdot 1 = c

2 sec (x) - c = 0

Move

c

to the right side

2 sec (x) - c = 0

Add

c

to both sides

2 sec (x) - c + c = 0 + c

Simplify

2 sec (x) = c

2 sec (x) = c

Divide both sides by

2

2 sec (x) = c

Divide both sides by

2

\frac{2 sec ( x )}{2} = \frac{c}{2}

Simplify

sec (x) = \frac{c}{2}

sec (x) = \frac{c}{2}

Apply trig inverse properties

sec (x) = \frac{c}{2}

General solutions for

sec (x) = \frac{c}{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (\frac{c}{2}) + 2 πn, x = - arcsec (\frac{c}{2}) + 2 πn

x = arcsec (\frac{c}{2}) + 2 πn, x = - arcsec (\frac{c}{2}) + 2 πn

0.25 \cdot cos (x) = - 0.249

cos (x) sin (x) cos (x) = 1 - sin^{2} (x)

sin (θ) = 1 - (1 - (cos (θ) - 1)^{2})

5 cos (x) - 3 sin (x) = 4

14.79 m = (\frac{7.6 1 _{0}^{2} sin ( 2 θ _{0} )}{g})