Question

if sin A + sin square A =1 prove that cos squareA + cos power 4 A=1

Answer 1

Given:

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

To Prove:

$\rm\cos^2A+\cos^4A = 1$

$\rule{320}{1}$

We will use only one identity:

$\rm 1-\sin^2\theta = \cos^2\theta$

Proof:

$\rm\textsf{Consider }\sin A + \sin^2A=1 \\\\\\ \implies \rm \sin A=1-\sin^2A\\\\\\ \implies \rm\sin A=\cos^2A \\\\\\ \textsf{Squaring both sides} \\\\\\ \implies\rm \sin^2 A=\cos^4 A \\\\\\ \implies \rm 1-\cos^2 = \cos^4A\\\\\\ \implies \rm 1 = \cos^2 A+\cos^4 A \\\\\\ \implies \boxed{\rm \cos^2A+\cos^4A=1} \\\\\\ \mathcal{HENCE\ \ PROVED}$

Answer 2

Answer:

$\bold\red{Given\::}$

• $\rm sinA + \sin^2 A = 1$

$\bold\green{To\:prove\::}$

• $\rm \cos^2 A + \cos^4 A = 1$

Formula used:-

${\boxed{\bold{{cos}^{2} \theta = 1 - {sin}^{2}\theta }}}$

$\rightarrow\rm sinA + \sin^2 A = 1 \\\\\rightarrow\rm sinA = 1 - \sin^2 A \\\\\rightarrow\rm sinA = \cos^2A$

$\star\tt By\: squaring \:both\:sides$

$\rightarrow\rm \sin^2 A = \cos^4 A \\\\\rightarrow\rm 1 - \cos^2 A = \cos^4 A \\\\\rightarrow\rm \cos^2 A + \cos^4 A = 1 \: \: \: [Proved]$

$\therefore\bold{\underline{{cos}^{2}A + {cos}^{4} A = 1 \: \: \: [Proved]}}$