Question

$if \: { \cot }^{4} a - { \cot }^{2}a = 1 \\ then \\ prove \ it \: \\ { \cos }^{4} a + { \cos}^{2}a = 1 \:$
this is the math of trigonometry.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: {cot}^{4}a - {cot}^{2}a = 1$

$\rm \: {cot}^{2}a\bigg({cot}^{2}a - 1\bigg) = 1$

can be further rewritten as

$\rm \: \dfrac{ {cos}^{2} a}{ {sin}^{2} a} \bigg(\dfrac{ {cos}^{2} a}{ {sin}^{2}a} - 1 \bigg) = 1$

$\rm \: \dfrac{ {cos}^{2} a}{ {sin}^{2} a} \bigg(\dfrac{ {cos}^{2} a - {sin}^{2}a}{ {sin}^{2}a} \bigg) = 1$

$\rm \: {cos}^{2}a( {cos}^{2}a - {sin}^{2}a) = {( {sin}^{2} a)}^{2}$

can be further rewritten as

$\rm \: {cos}^{2}a\bigg[{cos}^{2}a - (1 - {cos}^{2}a)\bigg] = {(1 - {cos}^{2} a)}^{2}$

$\rm \: {cos}^{2}a\bigg[{cos}^{2}a -1 + {cos}^{2}a\bigg] = 1 + {cos}^{4}a - {2cos}^{2} a$

$\rm \: {cos}^{2}a\bigg[2{cos}^{2}a -1 \bigg] = 1 + {cos}^{4}a - {2cos}^{2} a$

$\rm \: 2{cos}^{4}a - {cos}^{2}a = 1 + {cos}^{4}a - {2cos}^{2} a$

$\rm \: 2{cos}^{4}a - {cos}^{2}a - {cos}^{4}a + {2cos}^{2} a = 1$

$\rm\implies \:\rm \: {cos}^{4}a + {cos}^{2}a = 1$

$\rule{190pt}{2pt}$

Formulae Used :-

$\boxed{ \rm{ \:cotx = \frac{cosx}{sinx} \: }}$

$\boxed{ \rm{ \: {sin}^{2}x + {cos}^{2}x = 1 \: }}$

$\boxed{ \rm{ \: {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{sin(90 \degree - x) = cosx}\\ \\ \bigstar \: \bf{cos(90 \degree - x) = sinx}\\ \\ \bigstar \: \bf{tan(90 \degree - x) = cotx}\\ \\ \bigstar \: \bf{cot(90 \degree - x) = tanx}\\ \\ \bigstar \: \bf{cosec(90 \degree - x) = secx}\\ \\ \bigstar \: \bf{sec(90 \degree - x) = cosecx}\\ \\ \bigstar \: \bf{ {sin}^{2}x + {cos}^{2}x = 1 } \\ \\ \bigstar \: \bf{ {sec}^{2}x - {tan}^{2}x = 1 }\\ \\ \bigstar \: \bf{ {cosec}^{2}x - {cot}^{2}x = 1 }\\\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$

Answer 2

Answer:

your solution is as follows

pls mark it as brainliest

Step-by-step explanation:

$to \: prove \: that : \\ cos {}^{4} a + cos {}^{2} a = 1 \\ \\ given \: that \\ cot {}^{4} a - cot {}^{2} a = 1 \\ \\ cot {}^{2} a(cot {}^{2} a - 1) = 1 \\ \\ cot {}^{2}a - 1 = \frac{1}{cot {}^{2} a} \\ \\ cot {}^{2} a = tan {}^{2} a + 1 \\ \\ \frac{cos {}^{2} }{sin {}^{2}a } = sec {}^{2} a \\ \\ \frac{cos {}^{2} a}{sin {}^{2}a } = \frac{1}{cos {}^{2} a} \\ \\ cos {}^{4}a = sin {}^{2} a \\ \\ cos {}^{4} a = 1 - cos {}^{2} a \\ \\ cos {}^{4} a + cos {}^{2} a = 1$