# How do you solve (1+tanA)^2 + (1+cotA)^2 = (secA+cosecA)^2?

Hence, the values ofA satisfying the equation

${\left(1 + \tan A\right)}^{2} + {\left(1 + \cot A\right)}^{2} = {\left(\sec A + \csc A\right)}^{2}$ are,

$A = \ldots \ldots , \frac{- 7 \pi}{2} , - 3 \pi , \frac{- 5 \pi}{2} , - 2 \pi , \frac{- 3 \pi}{2} , - \pi , - \frac{\pi}{2} , 0 , \frac{\pi}{2} , \pi , \frac{3 \pi}{2} , 2 \pi , \frac{5 \pi}{2} , 3 \pi , \frac{7 \pi}{2} , \ldots \ldots$

#### Explanation:

${\left(1 + \tan A\right)}^{2} + {\left(1 + \cot A\right)}^{2} = {\left(\sec A + \csc A\right)}^{2}$

$\tan A = \sin \frac{A}{\cos} A$

$\cot A = \cos \frac{A}{\sin} A$

$\sec A = \frac{1}{\cos} A$

$\csc A = \frac{1}{\sin} A$

${\left(1 + \sin \frac{A}{\cos} A\right)}^{2} + {\left(1 + \cos \frac{A}{\sin} A\right)}^{2} = {\left(\frac{1}{\cos} A + \frac{1}{\sin} A\right)}^{2}$

${\left(\cos A + \sin A\right)}^{2} / {\cos}^{2} A + {\left(\sin A + \cos A\right)}^{2} / {\sin}^{2} A = {\left(\sin A + \cos A\right)}^{2} / \left(\sin A \cos A\right)$

${\left(\cos A + \sin A\right)}^{2} \left(\frac{1}{\cos} ^ 2 A + \frac{1}{\sin} ^ 2 A\right) = {\left(\cos A + \sin A\right)}^{2} \left(\frac{1}{\sin A \cos A}\right)$

$\frac{{\sin}^{2} A + {\cos}^{2} A}{{\sin}^{2} A {\cos}^{2} A} = \frac{1}{\sin A \cos A}$

$\frac{1}{\sin A \cos A} ^ 2 = \frac{1}{\sin A \cos A}$

${\left(\sin A \cos A\right)}^{2} = \sin A \cos A$

${\left(\sin A \cos A\right)}^{2} - \sin A \cos A = 0$

$\sin A \cos A \left(\sin A \cos A - 1\right) = 0$

$\sin A = 0 , \cos A = 0 , \sin A \cos A - 1 = 0$

$\sin A = 0 \to A = 0 , \pi , 2 \pi , 3 \pi , \ldots$
$\cos A = 0 \to A = \frac{\pi}{2} , \frac{3 \pi}{2} , \frac{5 \pi}{2} , \frac{7 \pi}{2} , \ldots \ldots$
$\sin A \cos A = 0 \to \frac{1}{2} \sin 2 A = 0 \to \sin 2 A = 0$
$2 A = 0 , \pi , 2 \pi , 3 \pi , \ldots$
$A = 0 , \frac{\pi}{2} , \pi , \frac{3 \pi}{2} , 2 \pi , \frac{5 \pi}{2} , 3 \pi , \frac{7 \pi}{2} , \ldots \ldots$

Hence, the values ofA satisfying the equation

${\left(1 + \tan A\right)}^{2} + {\left(1 + \cot A\right)}^{2} = {\left(\sec A + \csc A\right)}^{2}$ are,

$A = \ldots \ldots , \frac{- 7 \pi}{2} , - 3 \pi , \frac{- 5 \pi}{2} , - 2 \pi , \frac{- 3 \pi}{2} , - \pi , - \frac{\pi}{2} , 0 , \frac{\pi}{2} , \pi , \frac{3 \pi}{2} , 2 \pi , \frac{5 \pi}{2} , 3 \pi , \frac{7 \pi}{2} , \ldots \ldots$