What is the general solution for sec^2(x)+cot^2(x)=csc^2(x) ?

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

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sec^2(x)+cot^2(x)=csc^2(x)

sec^{2} (x) + cot^{2} (x) = csc^{2} (x)

No Solution for x \in R

Solution steps

sec^{2} (x) + cot^{2} (x) = csc^{2} (x)

Subtract

csc^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Use the Pythagorean identity:

csc^{2} (x) = 1 + cot^{2} (x)

csc^{2} (x) - cot^{2} (x) = 1

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

tan^{2} (x) = 0

tan^{2} (x) = 0

Apply rule

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

tan (x) = 0

General solutions for

tan (x) = 0

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 = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

Since the equation is undefined for:

πn

No Solution for x \in R

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\frac{( sin ^{2} ( x ) )}{( 1 - cos ( x ) )} = 1.23

- 1 = tan (2 (0)^{3} + C)

sin (x) = \frac{1}{2}, 0 < x < \frac{π}{2}

sin (2 x) = cos (8 x)

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