My Ist day on brilliant. & I am searching for friends.

$ \\ Given\quad \sin { \theta } +\sin ^{ 2 }{ \theta } =1\\ find\quad \cos ^{ 12 }{ \theta } +3\cos ^{ 10 }{ \theta } +3\cos ^{ 8 }{ \theta } +\cos ^{ 6 }{ \theta } \quad -1\quad .$

This was my first question in brilliant ( excluding the question: next number in the sequence 2,4,8.. during account creation).

$Since\quad value\quad is\quad given\quad in\quad terms\quad of\quad sin\theta \quad I\quad thought\\ it\quad might\quad be\quad beneficial\quad to\quad convert\quad terms\quad of\quad cos\theta \quad in\quad \\ sin\theta \quad moreover\quad its\quad apparent\quad that\quad it\quad can\quad be\quad done\quad easily.\quad \\ After\quad substituting(\cos ^{ 2 }{ \theta } =\sin { \theta } )\quad I\quad got\quad these\quad terms:\\ \\ \sin ^{ 3 }{ \theta } +\sin ^{ 6 }{ \theta } +3\sin ^{ 4 }{ \theta } +3\sin ^{ 5 }{ \theta } -1\\ \\ Which\quad on\quad rearranging\quad gave\quad these:\\ \\ \Rightarrow \sin ^{ 3 }{ \theta } +\sin ^{ 6 }{ \theta } +3\sin ^{ 3 }{ \theta } (\sin ^{ 2 }{ \theta } +\sin { \theta } )-1\\ \\ I\quad finally\quad shot\quad the\quad problem\quad by\quad this:\\ \Rightarrow { (\sin { \theta } +\sin ^{ 2 }{ \theta } ) }^{ 3 }-1=0.\\ \\ I\quad used\quad to\quad face\quad (to\quad be\quad honest,I\quad still\quad face)\quad difficulty\quad while\quad \\ attempting\quad 'trigo'\quad problems.They\quad appear\quad to\quad me\quad so\quad absurd.\\ 'What\quad should\quad we\quad do\quad here?'\\ 'Why\quad would\quad anyone\quad do\quad that?'\\ 'OK!\quad good\quad solution\quad there\quad but\quad how\quad to\quad arrive\quad at\quad it?'\\ These\quad are\quad the\quad family\quad of\quad questions\quad clouding\quad over\quad my\\ mind.\quad So,\quad I\quad (honestly\quad try)\quad to\quad reason\quad out\quad my\quad steps.\\ \\ I\quad am\quad not\quad good\quad at\quad Mathematics.\quad But,\quad I\quad like\quad Maths.\quad \\ Here\quad in\quad brilliant.org,\quad there\quad are\quad very\quad talented\quad problem\quad solvers.\\ I\quad know\quad it\quad because\quad I\quad had\quad spent\quad some\quad time\quad with\quad my\quad friend,\quad \\ in\quad his\quad profile,\quad tackling\quad problems\quad and\quad viewing\\ solutions\quad and\quad reading\quad notes.\quad Then\quad it\quad allured\quad me\quad to\quad join\quad it.\\ \\ I\quad don't\quad have\quad collections\quad of\quad nut-cracking\quad problems\quad neither\quad \\ the\quad ability\quad to\quad shoot\quad those\quad type\quad of\quad questions\quad but\quad still\quad if\\ some\quad are\quad benefited\\ in\quad any\quad kind\quad through\quad me\quad \\ \\ I\quad would\quad be\quad glad.\\ \\$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

My Ist day on brilliant. & I am searching for friends.

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