To verify the given trigonometric identity:
[tex]\[ \frac{1+\tan^2 \theta}{1+\cot^2 \theta}=\left(\frac{1-\tan \theta}{1-\cot \theta}\right)^2 \][/tex]
we will simplify both sides of the equation step-by-step and check if they are equal.
First, let's simplify the left side.
The left side of the equation is:
[tex]\[ \frac{1 + \tan^2 \theta}{1 + \cot^2 \theta} \][/tex]
Recall that [tex]\(\cot \theta\)[/tex] is the reciprocal of [tex]\(\tan \theta\)[/tex]:
[tex]\[ \cot \theta = \frac{1}{\tan \theta} \][/tex]
Thus, [tex]\(\cot^2 \theta\)[/tex] is:
[tex]\[ \cot^2 \theta = \left(\frac{1}{\tan \theta}\right)^2 = \frac{1}{\tan^2 \theta} \][/tex]
Now substitute [tex]\(\cot^2 \theta\)[/tex] into the equation:
[tex]\[ \frac{1 + \tan^2 \theta}{1 + \frac{1}{\tan^2 \theta}} \][/tex]
To simplify this, note that the denominator can be expressed with a common denominator:
[tex]\[ 1 + \frac{1}{\tan^2 \theta} = \frac{\tan^2 \theta + 1}{\tan^2 \theta} \][/tex]
Thus, the left side becomes:
[tex]\[ \frac{1 + \tan^2 \theta}{\frac{\tan^2 \theta + 1}{\tan^2 \theta}} = \frac{(1 + \tan^2 \theta) \cdot \tan^2 \theta}{\tan^2 \theta + 1} \][/tex]
Notice that [tex]\((1 + \tan^2 \theta) \cdot \tan^2 \theta\)[/tex] simplifies to [tex]\(\tan^2 \theta (1 + \tan^2 \theta)\)[/tex].
Therefore, the left side simplifies to:
[tex]\[ \tan^2 \theta \][/tex]
Next, let's simplify the right side.
The right side of the equation is:
[tex]\[ \left(\frac{1 - \tan \theta}{1 - \cot \theta}\right)^2 \][/tex]
Using [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex], we get:
[tex]\[ \left(\frac{1 - \tan \theta}{1 - \frac{1}{\tan \theta}}\right)^2 \][/tex]
To simplify, express the denominator with a common denominator:
[tex]\[ 1 - \frac{1}{\tan \theta} = \frac{\tan \theta - 1}{\tan \theta} \][/tex]
Thus, the right side becomes:
[tex]\[ \left(\frac{1 - \tan \theta}{\frac{\tan \theta - 1}{\tan \theta}}\right)^2 = \left((1 - \tan \theta) \cdot \frac{\tan \theta}{\tan \theta - 1}\right)^2 \][/tex]
Notice that [tex]\( (1 - \tan \theta) \)[/tex] and [tex]\( (\tan \theta - 1) \)[/tex] are opposites, so:
[tex]\[ (1 - \tan \theta)( \tan \theta) / (\tan \theta - 1)
= (\tan \theta (1 - \tan \theta)) / (-(1 - \tan \theta))
= - \tan^2 \theta \][/tex]
Therefore, the right side simplifies to:
[tex]\[ \left(-\tan \theta \right)^2 \][/tex]
And finally:
[tex]\[ \left(\tan^2 \theta \right) \][/tex]
Both sides, the left and the right, simplify to [tex]\(\tan^2 \theta \)[/tex],
Thus, the given identity:
[tex]\[ \frac{1+\tan^2 \theta}{1+\cot^2 \theta}=\left(\frac{1-\tan \theta}{1-\cot \theta}\right)^2 \][/tex]
is indeed verified to be true.
