Question

.
$if \: sin \alpha + \sin {}^{2} \alpha = 1 \: \\ then \\ \cos {}^{12} ( \alpha ) + 3 \cos {}^{10} ( \alpha ) + 3 \cos {}^{8} ( \alpha ) + \cos {}^{6} ( \alpha ) + \cos {}^{4} ( \alpha ) + 2 \cos {}^{2} ( \alpha ) - 2$
please give the correct answer
correct answer will mark brainlist
​

Answer 1

Appropriate Question

If

$\rm \: sin \alpha + {sin}^{2}\alpha = 1$

then

$\rm \: {cos}^{12}\alpha + 3 {cos}^{10}\alpha + 3 {cos}^{8}\alpha + {cos}^{6}\alpha + 2 {cos}^{4}\alpha + {2cos}^{2}\alpha - 2 =$

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

Given that,

$\rm \: sin\alpha + {sin}^{2}\alpha = 1$

$\rm \: sin\alpha = 1 - {sin}^{2}\alpha$

$\rm\implies \:sin\alpha = {cos}^{2}\alpha - - - (1)$

Now, Consider

$\rm \: {cos}^{12}\alpha + 3 {cos}^{10}\alpha + 3 {cos}^{8}\alpha + {cos}^{6}\alpha + 2 {cos}^{4}\alpha + {2cos}^{2}\alpha - 2$

can be rewritten as using equation (1),

$\rm \: = \: {sin}^{6}\alpha + 3 {sin}^{5}\alpha + 3 {sin}^{4}\alpha + {sin}^{3}\alpha + 2 {sin}^{2}\alpha + 2sin\alpha - 2$

$\rm \: = \: {sin}^{3}\alpha ( {sin}^{3}\alpha + 3 {sin}^{2}\alpha + 3sin\alpha + 1) + 2( {sin}^{2}\alpha + sin\alpha - 1)$

We know,

$\boxed{\tt{ {x}^{3} + {3x}^{2} + 3x + 1 = {(x + 1)}^{3} \: }}$

and also given that

$\boxed{\tt{ sin\alpha + {sin}^{2}\alpha = 1 \: }}$

So, using this, we get

$\rm \: = \: {sin}^{3}\alpha {(sin\alpha + 1)}^{3} + 2(1 - 1)$

$\rm \: = \: {\bigg(sin\alpha (sin\alpha + 1)\bigg) }^{3} + 2 \times 0$

$\rm \: = \: {\bigg( {sin}^{2}\alpha + sin\alpha \bigg) }^{3} + 0$

$\rm \: = \: {\bigg(1\bigg) }^{3}$

$\rm \: = \: 1$

Hence,

$\boxed{\tt{ \rm \: {cos}^{12}\alpha + 3 {cos}^{10}\alpha + 3 {cos}^{8}\alpha + {cos}^{6}\alpha + 2 {cos}^{4}\alpha + {2cos}^{2}\alpha - 2 = 1}}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1