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

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

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(sec^2(A)+cot^2(A))/(csc^2(A))=sec^2(A)

Solution

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} = sec^{2} (A)

Solution

No Solution for A \in R

Solution steps

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} = sec^{2} (A)

Subtract

sec^{2} (A)

from both sides

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} - sec^{2} (A) = 0

Simplify

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} - sec^{2} (A) : \frac{sec ^{2} ( A ) + cot ^{2} ( A ) - sec ^{2} ( A ) csc ^{2} ( A )}{csc ^{2} ( A )}

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} - sec^{2} (A)

Convert element to fraction:

sec^{2} (A) = \frac{sec ^{2} ( A ) csc ^{2} ( A )}{csc ^{2} ( A )}

= \frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} - \frac{sec ^{2} ( A ) csc ^{2} ( A )}{csc ^{2} ( A )}

Since the denominators are equal, combine the fractions:

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

= \frac{sec ^{2} ( A ) + cot ^{2} ( A ) - sec ^{2} ( A ) csc ^{2} ( A )}{csc ^{2} ( A )}

\frac{sec ^{2} ( A ) + cot ^{2} ( A ) - sec ^{2} ( A ) csc ^{2} ( A )}{csc ^{2} ( A )} = 0

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= cot^{2} (A) + sec^{2} (A) - (1 + cot^{2} (A)) sec^{2} (A)

Simplify

cot^{2} (A) + sec^{2} (A) - (1 + cot^{2} (A)) sec^{2} (A) : cot^{2} (A) - sec^{2} (A) cot^{2} (A)

cot^{2} (A) + sec^{2} (A) - (1 + cot^{2} (A)) sec^{2} (A)

= cot^{2} (A) + sec^{2} (A) - sec^{2} (A) (1 + cot^{2} (A))

Expand

- sec^{2} (A) (1 + cot^{2} (A)) : - sec^{2} (A) - sec^{2} (A) cot^{2} (A)

- sec^{2} (A) (1 + cot^{2} (A))

Apply the distributive law:

a (b + c) = ab + a c

a = - sec^{2} (A), b = 1, c = cot^{2} (A)

= - sec^{2} (A) \cdot 1 + (- sec^{2} (A)) cot^{2} (A)

Apply minus-plus rules

+ (- a) = - a

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

Multiply:

1 \cdot sec^{2} (A) = sec^{2} (A)

= - sec^{2} (A) - sec^{2} (A) cot^{2} (A)

= cot^{2} (A) + sec^{2} (A) - sec^{2} (A) - sec^{2} (A) cot^{2} (A)

Add similar elements:

sec^{2} (A) - sec^{2} (A) = 0

= cot^{2} (A) - sec^{2} (A) cot^{2} (A)

= cot^{2} (A) - sec^{2} (A) cot^{2} (A)

cot^{2} (A) - cot^{2} (A) sec^{2} (A) = 0

Factor

cot^{2} (A) - cot^{2} (A) sec^{2} (A) : - cot^{2} (A) (sec (A) + 1) (sec (A) - 1)

cot^{2} (A) - cot^{2} (A) sec^{2} (A)

Factor out common term

- cot^{2} (A)

= - cot^{2} (A) (- 1 + sec^{2} (A))

Factor

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

sec^{2} (A) - 1

Rewrite

1

1^{2}

= sec^{2} (A) - 1^{2}

Apply Difference of Two Squares Formula:

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

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

= (sec (A) + 1) (sec (A) - 1)

= - cot^{2} (A) (sec (A) + 1) (sec (A) - 1)

- cot^{2} (A) (sec (A) + 1) (sec (A) - 1) = 0

Solving each part separately

cot^{2} (A) = 0 or sec (A) + 1 = 0 or sec (A) - 1 = 0

cot^{2} (A) = 0 : A = \frac{π}{2} + πn

cot^{2} (A) = 0

Apply rule

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

cot (A) = 0

General solutions for

cot (A) = 0

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

A = \frac{π}{2} + πn

A = \frac{π}{2} + πn

sec (A) + 1 = 0 : A = π + 2 πn

sec (A) + 1 = 0

Move

1

to the right side

sec (A) + 1 = 0

Subtract

1

from both sides

sec (A) + 1 - 1 = 0 - 1

Simplify

sec (A) = - 1

sec (A) = - 1

General solutions for

sec (A) = - 1

sec (x)

periodicity table with

2 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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

A = π + 2 πn

A = π + 2 πn

sec (A) - 1 = 0 : A = 2 πn

sec (A) - 1 = 0

Move

1

to the right side

sec (A) - 1 = 0

Add

1

to both sides

sec (A) - 1 + 1 = 0 + 1

Simplify

sec (A) = 1

sec (A) = 1

General solutions for

sec (A) = 1

sec (x)

periodicity table with

2 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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

A = 0 + 2 πn

A = 0 + 2 πn

Solve

A = 0 + 2 πn : A = 2 πn

A = 0 + 2 πn

0 + 2 πn = 2 πn

A = 2 πn

A = 2 πn

Combine all the solutions

A = \frac{π}{2} + πn, A = π + 2 πn, A = 2 πn

Since the equation is undefined for:

\frac{π}{2} + πn, π + 2 πn, 2 πn

No Solution for A \in R

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Frequently Asked Questions (FAQ)

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