What is the general solution for ((1-tan^2(a)))/((tan(a)))=2cot^2(a) ?

The general solution for ((1-tan^2(a)))/((tan(a)))=2cot^2(a) is a=2.15228…+pin

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((1-tan^2(a)))/((tan(a)))=2cot^2(a)

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

a = 2.15228 \dots + πn

Solution steps

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

Subtract

2 cot^{2} (a)

from both sides

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

Simplify

\frac{1 - tan ^{2} ( a )}{tan ( a )} - 2 cot^{2} (a) : \frac{1 - tan ^{2} ( a ) - 2 cot ^{2} ( a ) tan ( a )}{tan ( a )}

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

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

1 - tan^{2} (a) - 2 cot^{2} (a) tan (a) = 0

Rewrite using trig identities

1 - \frac{1}{cot ^{2} ( a )} - 2 cot (a) = 0

Solve by substitution

cot (a) \approx - 0.65729 \dots

cot (a) = - 0.65729 \dots : a = arccot (- 0.65729 \dots) + πn

Combine all the solutions

a = arccot (- 0.65729 \dots) + πn

Show solutions in decimal form

a = 2.15228 \dots + πn

cos (6 x) = cos (2 x)

cot (x) cos (x) - sin (x) = 1

cos^{4} (x) + cos^{3} (x) - 2 = 0

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

\frac{sin ^{2} ( x ) - 2 cos ( x ) + 1}{4} = 0

What is the general solution for ((1-tan^2(a)))/((tan(a)))=2cot^2(a) ?
The general solution for ((1-tan^2(a)))/((tan(a)))=2cot^2(a) is a=2.15228…+pin