What is the general solution for cos(x)((cos(x))/(sin(x)))+sin(x)=sec(x) ?

The general solution for cos(x)((cos(x))/(sin(x)))+sin(x)=sec(x) is x= pi/4+pin

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cos(x)((cos(x))/(sin(x)))+sin(x)=sec(x)

Solution

cos (x) (\frac{cos ( x )}{sin ( x )}) + sin (x) = sec (x)

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

cos (x) (\frac{cos ( x )}{sin ( x )}) + sin (x) = sec (x)

Subtract

sec (x)

from both sides

\frac{cos ^{2} ( x )}{sin ( x )} + sin (x) - sec (x) = 0

Simplify

\frac{cos ^{2} ( x )}{sin ( x )} + sin (x) - sec (x) : \frac{cos ^{2} ( x ) + sin ^{2} ( x ) - sec ( x ) sin ( x )}{sin ( x )}

\frac{cos ^{2} ( x )}{sin ( x )} + sin (x) - sec (x)

Convert element to fraction:

sin (x) = \frac{sin ( x ) sin ( x )}{sin ( x )}, sec (x) = \frac{sec ( x ) sin ( x )}{sin ( x )}

= \frac{cos ^{2} ( x )}{sin ( x )} + \frac{sin ( x ) sin ( x )}{sin ( x )} - \frac{sec ( x ) sin ( x )}{sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ^{2} ( x ) + sin ( x ) sin ( x ) - sec ( x ) sin ( x )}{sin ( x )}

cos^{2} (x) + sin (x) sin (x) - sec (x) sin (x) = cos^{2} (x) + sin^{2} (x) - sec (x) sin (x)

cos^{2} (x) + sin (x) sin (x) - sec (x) sin (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

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

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

= cos^{2} (x) + sin^{2} (x) - sec (x) sin (x)

= \frac{cos ^{2} ( x ) + sin ^{2} ( x ) - sec ( x ) sin ( x )}{sin ( x )}

\frac{cos ^{2} ( x ) + sin ^{2} ( x ) - sec ( x ) sin ( x )}{sin ( x )} = 0

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

cos^{2} (x) + sin^{2} (x) - sec (x) sin (x) = 0

Rewrite using trig identities

- sec (x) sin (x) + 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Express with sin, cos

sec (x) sin (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( x )} sin (x)

Simplify

\frac{1}{cos ( x )} sin (x) : \frac{sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} sin (x)

Multiply fractions:

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

= \frac{1 sin ( x )}{cos ( x )}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{cos ( x )}

= \frac{sin ( x )}{cos ( x )}

= \frac{sin ( x )}{cos ( x )}

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

= tan (x)

= 1 - tan (x)

1 - tan (x) = 0

Move

1

to the right side

1 - tan (x) = 0

Subtract

1

from both sides

1 - tan (x) - 1 = 0 - 1

Simplify

- tan (x) = - 1

- tan (x) = - 1

Divide both sides by

- 1

- tan (x) = - 1

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{- 1}{- 1}

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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cos (2 t) - sin (t) = 0.5

0=sin^2(x)

0 = sin^{2} (x)

sec(40+2θ)=csc(15)

sec (4 0^{\circ} + 2 θ) = csc (1 5^{\circ})

arcsin(x)-arccos(x)=arcsin(1/2)

arcsin (x) - arccos (x) = arcsin (\frac{1}{2})

cos^2(x)-cos(x)=sin^2(x)

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

Frequently Asked Questions (FAQ)

What is the general solution for cos(x)((cos(x))/(sin(x)))+sin(x)=sec(x) ?
The general solution for cos(x)((cos(x))/(sin(x)))+sin(x)=sec(x) is x= pi/4+pin