Question

If cosA + cos^2A = 1, then find the value of sin^2A + sin^4A​

Answer 1

❒ Question :

$\large \sf If \: Cos \: A + Cos^{2} \: A = 1, \\ \large \sf then \: find \: the \: value \: of \:Sin^{2} \: A + Sin^{4} \: A$

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❒ Solution :

$\bf \: The \: value \: of \:Sin^{2} \: A + Sin^{4} \: A = \red 1$

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$⇝ \: \sf Cos \: A + Cos^{2} \: A = 1$

$⟹ \sf \:Sin^{2} \: A + Sin^{4} \: A \:$

$⟹ \sf \:(1 - Cos^{2} \: A) + (1 - Cos^{2} \: A) ^{2}$

$⟹ \sf \:(1 -(1 - Cos \: A) \: + \: (1 -(1 - Cos \: A) ^{2}$

$⟹ \sf \: 0+ Cos \: A \: + \: (0 + Cos \: A) ^{2}$

$\red{⟹ \sf \: Cos \: A \: + \: (Cos \: A) ^{2} = 1}$

Hence,

$∴ \sf \: The \: value \: of \:Sin^{2} \: A + Sin^{4} \: A = 1$

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❒ Step-by-step Explanation :

★ Given :-

$\sf Cos \: A + Cos^{2} \: A = 1$

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★ To Find :-

$\sf \: The \: value \: of \:Sin^{2} \: A + Sin^{4} \: A$

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★ Used Identity :-

$\bf⇢ \: Sin^{2}A + Cos^{2} A = 1 \\ \: \: \: \: \: \: \: \: \bf \red{ Sin^{2}A = 1 - Cos^{2} A }\: \: \: \: \: \: \: \: \: \sf(by \: transposing)$

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★ Explanation :-

Here,

$⇝ \: \sf Cos \: A + Cos^{2} \: A = 1$

$⇝ \sf \:Sin^{2} \: A + Sin^{4} \: A \: = \: ?$

Now,

$⟹ \sf \:Sin^{2} \: A + Sin^{4} \: A \:$

We know that,

$\bf⟹ \: Sin^{2}A = 1 - Cos^{2} A$

So,

$⟹ \sf \:(1 - Cos^{2} \: A) + (1 - Cos^{2} \: A) ^{2} \:$

We can write it as :

$⟹ \sf \:(1 -(1 - Cos \: A) \: + \: (1 -(1 - Cos \: A) ^{2} \:$

And,

$⟹ \sf \: 1 -1 + Cos \: A \: + \: (1 -1 + Cos \: A) ^{2} \:$

$⟹ \sf \: 0+ Cos \: A \: + \: (0 + Cos \: A) ^{2} \:$

$⟹ \sf \: Cos \: A \: + \: (Cos \: A) ^{2}$

So, according to the question we get,

$\red{⟹ \: \bf Cos \: A + Cos^{2} \: A = 1 }$

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Answer 2

Step-by-step explanation:

cosA+cos²A=1

cosA=1−cos² A

cosA=sin²A

Hence

sin 4A+sin²A

cos² A+cosA

=1