What is the general solution for (tan^2(x)(1+cot^2(x)))/(sec(x))=cos(x) ?

The general solution for (tan^2(x)(1+cot^2(x)))/(sec(x))=cos(x) is No Solution for x\in\mathbb{R}

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(tan^2(x)(1+cot^2(x)))/(sec(x))=cos(x)

\frac{tan ^{2} ( x ) ( 1 + cot ^{2} ( x ) )}{sec ( x )} = cos (x)

No Solution for x \in R

Solution steps

\frac{tan ^{2} ( x ) ( 1 + cot ^{2} ( x ) )}{sec ( x )} = cos (x)

Subtract

cos (x)

from both sides

\frac{tan ^{2} ( x ) ( 1 + cot ^{2} ( x ) )}{sec ( x )} - cos (x) = 0

Simplify

\frac{tan ^{2} ( x ) ( 1 + cot ^{2} ( x ) )}{sec ( x )} - cos (x) : \frac{tan ^{2} ( x ) ( 1 + cot ^{2} ( x ) ) - cos ( x ) sec ( x )}{sec ( x )}

\frac{tan ^{2} ( x ) ( 1 + cot ^{2} ( x ) ) - cos ( x ) sec ( x )}{sec ( x )} = 0

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

tan^{2} (x) (1 + cot^{2} (x)) - cos (x) sec (x) = 0

Rewrite using trig identities

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

Move

1

to the right side

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

Move

1

to the right side

tan^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

tan (x) = 0

General solutions for

tan (x) = 0

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = πn

Since the equation is undefined for:

πn

No Solution for x \in R

4 cos^{2} (x) tan (x) - tan (x) = 0

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

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

csc (θ) = (1.201)

3 cos (θ) + 2 = 5 sec (θ), 0 \leq θ < 2 π

What is the general solution for (tan^2(x)(1+cot^2(x)))/(sec(x))=cos(x) ?
The general solution for (tan^2(x)(1+cot^2(x)))/(sec(x))=cos(x) is No Solution for x\in\mathbb{R}