If $ \cos A+\cos ^{2} A=1 $, prove that $ \sin ^{2} A+\sin ^{4} A=1 $

Given:

\( \cos A+\cos ^{2} A=1 \)

To do:

We have to prove that \( \sin ^{2} A+\sin ^{4} A=1 \).

Solution:

We know that,

$\sin^2 A+\cos^2 A=1$

$\operatorname{cosec}^2 A-\cot^2 A=1$

$\sec^2 A-\tan^2 A=1$

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

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

$\operatorname{cosec} A=\frac{1}{\sin A}$

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

Therefore,

$\cos A + \cos^2 A = 1$

$\Rightarrow \cos A = 1 - \cos^2 A$

$\Rightarrow \cos A = sin^2 A$

This implies,

$\sin^2 A + \sin^4 A = \sin^2 A + (\sin^2 A)^2$

$= \cos A + \cos^2 A$

$= 1$

Hence proved.      

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Updated on: 10-Oct-2022

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