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

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

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

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

No Solution for x \in R

Solution steps

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

Subtract

sec^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = sec^{2} (x), c = 1

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

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

Add similar elements:

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

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

Group like terms

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

2 - 2 = 0

= sec^{2} (x)

= sec^{2} (x)

= sec^{2} (x)

sec^{2} (x) = 0

Apply rule

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

sec (x) = 0

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

No Solution for x \in R

tan (x) = - \frac{2}{2}

cot (x) = 1.2

cos (\frac{3}{2} π - x) + sin (4 x) = 0

tan (x) = \frac{30}{100}

cos (θ) = - \frac{53}{8}

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