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

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

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

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

No Solution for x \in R

Solution steps

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

Subtract

sec (x)

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

- sec (x) (1 + sec (x)) + sec^{2} (x) : - sec (x)

- sec (x) (1 + sec (x)) + sec^{2} (x)

Expand

- sec (x) (1 + sec (x)) : - sec (x) - sec^{2} (x)

- sec (x) (1 + sec (x))

Apply the distributive law:

a (b + c) = ab + a c

a = - sec (x), b = 1, c = sec (x)

= - sec (x) \cdot 1 + (- sec (x)) sec (x)

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot sec (x) - sec (x) sec (x)

Simplify

- 1 \cdot sec (x) - sec (x) sec (x) : - sec (x) - sec^{2} (x)

- 1 \cdot sec (x) - sec (x) sec (x)

1 \cdot sec (x) = sec (x)

1 \cdot sec (x)

Multiply:

1 \cdot sec (x) = sec (x)

= sec (x)

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

sec (x) sec (x)

Apply exponent rule:

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

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

= sec^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sec^{2} (x)

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

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

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

Add similar elements:

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

= - sec (x)

= - sec (x)

- sec (x) = 0

Divide both sides by

- 1

- sec (x) = 0

Divide both sides by

- 1

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

Simplify

sec (x) = 0

sec (x) = 0

sec (x) \leq - 1 or sec (x) \geq 1

No Solution for x \in R

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

sin (5 x - 1) = \frac{4}{5}

sin^{2} (x) = \frac{1}{36}

tan (x) = 31

sin^{4} (x) + cos^{2} (x) = 2

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