What is the general solution for (cos(x))/(sec(x))-(sin(x))/(csc(x))=1 ?

The general solution for (cos(x))/(sec(x))-(sin(x))/(csc(x))=1 is No Solution for x\in\mathbb{R}

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(cos(x))/(sec(x))-(sin(x))/(csc(x))=1

\frac{cos ( x )}{sec ( x )} - \frac{sin ( x )}{csc ( x )} = 1

No Solution for x \in R

Solution steps

\frac{cos ( x )}{sec ( x )} - \frac{sin ( x )}{csc ( x )} = 1

Subtract

1

from both sides

\frac{cos ( x )}{sec ( x )} - \frac{sin ( x )}{csc ( x )} - 1 = 0

Simplify

\frac{cos ( x )}{sec ( x )} - \frac{sin ( x )}{csc ( x )} - 1 : \frac{cos ( x ) csc ( x ) - sin ( x ) sec ( x ) - sec ( x ) csc ( x )}{sec ( x ) csc ( x )}

\frac{cos ( x ) csc ( x ) - sin ( x ) sec ( x ) - sec ( x ) csc ( x )}{sec ( x ) csc ( x )} = 0

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

cos (x) csc (x) - sin (x) sec (x) - sec (x) csc (x) = 0

Express with sin, cos

\frac{- 1 + cos ^{2} ( x ) - sin ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

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

Rewrite using trig identities

- 1 + cos (2 x) = 0

Move

1

to the right side

cos (2 x) = 1

General solutions for

cos (2 x) = 1

2 x = 0 + 2 πn

Solve

2 x = 0 + 2 πn : x = πn

x = πn

Since the equation is undefined for:

πn

No Solution for x \in R

tan (x) = \frac{12}{35}

4 = sec (θ)

1 + 2 sin (x) = cos (2 x)

12 - sin (θ) = cos (2 θ)

2 cos (θ) = 3

What is the general solution for (cos(x))/(sec(x))-(sin(x))/(csc(x))=1 ?
The general solution for (cos(x))/(sec(x))-(sin(x))/(csc(x))=1 is No Solution for x\in\mathbb{R}