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# Prove the following trigonometric equation: $\dfrac{{{\tan }^{2}}A}{1+{{\tan }^{2}}A}+\dfrac{{{\cot }^{2}}A}{1+{{\cot }^{2}}A}=1$.

Last updated date: 12th Sep 2024
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Hint: The given question is related to trigonometric identities. Here use the formulae related to the relation between tangent and cotangent of an angle.

We need to prove that the value of $\dfrac{{{\tan }^{2}}A}{1+{{\tan }^{2}}A}+\dfrac{{{\cot }^{2}}A}{1+{{\cot }^{2}}A}$ is equal to $1$ .
First, we will consider the left-hand side of the equation which is given as $\dfrac{{{\tan }^{2}}A}{1+{{\tan }^{2}}A}+\dfrac{{{\cot }^{2}}A}{1+{{\cot }^{2}}A}$. We will take the LCM of the denominators and solve the fraction to determine the value of the left-hand side of the equation. The LCM of the denominators in the left-hand side of the equation is given as $\left( 1+{{\tan }^{2}}A \right)\left( 1+{{\cot }^{2}}A \right)$ .
So, the left-hand side of the equation becomes $\dfrac{{{\tan }^{2}}A\left( 1+{{\cot }^{2}}A \right)}{\left( 1+{{\tan }^{2}}A \right)\left( 1+{{\cot }^{2}}A \right)}+\dfrac{{{\cot }^{2}}A\left( 1+{{\tan }^{2}}A \right)}{\left( 1+{{\tan }^{2}}A \right)\left( 1+{{\cot }^{2}}A \right)}$.
$=\dfrac{{{\tan }^{2}}A\left( 1+{{\cot }^{2}}A \right)+{{\cot }^{2}}A\left( 1+{{\tan }^{2}}A \right)}{\left( 1+{{\tan }^{2}}A \right)\left( 1+{{\cot }^{2}}A \right)}$
$\dfrac{{{\tan }^{2}}A}{1+{{\tan }^{2}}A}+\dfrac{{{\cot }^{2}}A}{1+{{\cot }^{2}}A}=\dfrac{{{\tan }^{2}}A+{{\tan }^{2}}A{{\cot }^{2}}A+{{\cot }^{2}}A+{{\tan }^{2}}A{{\cot }^{2}}A}{1+{{\tan }^{2}}A+{{\tan }^{2}}A{{\cot }^{2}}A+{{\cot }^{2}}A}....(i)$
Now, we know that $\cot A=\dfrac{1}{\tan A}$ . So, $\cot A\tan A=1$ . Hence, ${{\cot }^{2}}A{{\tan }^{2}}A=1$.
Now, we will substitute ${{\cot }^{2}}A{{\tan }^{2}}A=1$ in equation $(i)$. So, we get: $\dfrac{{{\tan }^{2}}A}{1+{{\tan }^{2}}A}+\dfrac{{{\cot }^{2}}A}{1+{{\cot }^{2}}A}=\dfrac{{{\tan }^{2}}A+1+{{\cot }^{2}}A+1}{1+{{\tan }^{2}}A+{{\cot }^{2}}A+1}$. Clearly, the values of the numerator and the denominator are the same. So, the value of the fraction will be equal to $1$ . So, the value of $\dfrac{{{\tan }^{2}}A}{1+{{\tan }^{2}}A}+\dfrac{{{\cot }^{2}}A}{1+{{\cot }^{2}}A}$ will be equal to $1$, i.e.
$\dfrac{{{\tan }^{2}}A}{1+{{\tan }^{2}}A}+\dfrac{{{\cot }^{2}}A}{1+{{\cot }^{2}}A}=1$.
Hence, the value of the left-hand side of the equation is equal to $1$.
Now, we will consider the right-hand side of the equation. The value of the right-hand side of the equation is also equal to $1$.