Solve \(\frac{1}{\cot ^{2}a} \frac{1}{1 \tan ^{2}a}=\

Evaluate

\frac{1}{cot ^{2} ( a )} \times \frac{1}{1 \times tan ^{2} ( a )} = \frac{1}{1 - sin ^{2} ( a )} - \frac{1}{csc ^{2} ( a )}

Find the domain

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\frac{1}{cot ^{2} ( a )} \times \frac{1}{1 \times tan ^{2} ( a )} = \frac{1}{1 - sin ^{2} ( a )} - \frac{1}{csc ^{2} ( a )}, a \neq = \frac{kπ}{2}, k \in Z

Simplify

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1 = \frac{1}{1 - sin ^{2} ( a )} - \frac{1}{csc ^{2} ( a )}

Simplify

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1 = tan^{2} (a) + cos^{2} (a)

Swap the sides of the equation

tan^{2} (a) + cos^{2} (a) = 1

Rewrite the expression

(\frac{sin ( a )}{cos ( a )})^{2} + cos^{2} (a) = 1

Simplify

\frac{sin ^{2} ( a )}{cos ^{2} ( a )} + cos^{2} (a) = 1

Multiply both sides of the equation by LCD

(\frac{sin ^{2} ( a )}{cos ^{2} ( a )} + cos^{2} (a)) cos^{2} (a) = 1 \times cos^{2} (a)

Simplify the equation

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sin^{2} (a) + cos^{4} (a) = 1 \times cos^{2} (a)

Any expression multiplied by 1 remains the same

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

Move the expression to the left side

sin^{2} (a) + cos^{4} (a) - cos^{2} (a) = 0

Simplify

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2 sin^{2} (a) + cos^{4} (a) - 1 = 0

Simplify the equation using the Weierstrass substitution

2 (\frac{2 tan ( \frac{1}{2} a )}{1 + tan ^{2} ( \frac{1}{2} a )})^{2} + (\frac{1 - tan ^{2} ( \frac{1}{2} a )}{1 + tan ^{2} ( \frac{1}{2} a )})^{4} - 1 = 0

Solve using substitution

2 (\frac{2 t}{1 + t ^{2}})^{2} + (\frac{1 - t ^{2}}{1 + t ^{2}})^{4} - 1 = 0

Rearrange the terms

2 (\frac{2 t}{1 + t ^{2}})^{2} + (\frac{1 - t ^{2}}{1 + t ^{2}})^{4} - 1 = 0 a = 18 0^{\circ} + 36 0^{\circ} k, k \in Z

Calculate

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t = 0 a = 18 0^{\circ} + 36 0^{\circ} k, k \in Z

Substitute back

tan (\frac{1}{2} a) = 0 a = 18 0^{\circ} + 36 0^{\circ} k, k \in Z

Calculate

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a = 2 kπ, k \in Z a = 18 0^{\circ} + 36 0^{\circ} k, k \in Z

Check if x = π + 2 kπ, k \in Z is a solution

\frac{1}{cot ^{2} ( π + 2 kπ )} \times \frac{1}{1 \times tan ^{2} ( π + 2 kπ )} = \frac{1}{1 - sin ^{2} ( π + 2 kπ )} - \frac{1}{csc ^{2} ( π + 2 kπ )}

Calculate

\frac{1}{cot ^{2} ( π )} \times \frac{1}{1 \times tan ^{2} ( π )} = \frac{1}{1 - sin ^{2} ( π )} - \frac{1}{csc ^{2} ( π )}

Evaluate

Undefined

Since x = π + 2 kπ, k \in Z is not a solution,don’t include it

a = 2 kπ, k \in Z a = 18 0^{\circ} + 36 0^{\circ} k, k \in Z

Find the union

a = kπ, k \in Z

Check if the solution is in the defined range

a = kπ, k \in Z, a \neq = \frac{kπ}{2}, k \in Z

Solution

a \in \emptyset

Alternative Form

No solution

\(\frac{1}{\cot ^{2}a} \frac{1}{1 \tan ^{2}a}=\frac{1}{1 \sin ^{2}a} \frac{1}{\csc ^{2}a}\)