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Question

$s i n squared space alpha plus c o s squared space left parenthesis alpha plus beta right parenthesis plus 2 s i n space alpha s i n space beta c o s space left parenthesis alpha plus beta right parenthesis equals$

$s i n^{2} α$
$s i n^{2} β$
$c o s^{2} α$
$c o s^{2} β$

hint Hint:

In this question we have to take similar terms common and then use the formula of $cos open parentheses alpha plus beta close parentheses equals cos open parentheses alpha close parentheses cos open parentheses beta close parentheses minus sin open parentheses alpha close parentheses sin open parentheses beta close parentheses$ . The process of taking similar terms common will be done twice. We will use the identity $sin open parentheses theta close parentheses squared plus cos open parentheses theta close parentheses squared space equals 1$ to find the answer.

The correct answer is: $c o s squared space beta$

In this question we are given expression $s i n squared space alpha plus c o s squared space left parenthesis alpha plus beta right parenthesis plus 2 s i n space alpha s i n space beta c o s space left parenthesis alpha plus beta right parenthesis$ and we have to simplify it.
Step1: Using Trigonometric relation to simplify it.
$s i n squared space alpha plus c o s squared space left parenthesis alpha plus beta right parenthesis plus 2 s i n space alpha s i n space beta c o s space left parenthesis alpha plus beta right parenthesis$
In the last two terms of the expression we can take $cos open parentheses alpha plus beta close parentheses$ as common factor.
$s i n squared space alpha plus c o s space left parenthesis alpha plus beta right parenthesis left parenthesis c o s space left parenthesis alpha plus beta right parenthesis plus 2 s i n space alpha s i n space beta right parenthesis$
Step2: Using the formula $cos open parentheses alpha plus beta close parentheses equals cos open parentheses alpha close parentheses cos open parentheses beta close parentheses minus sin open parentheses alpha close parentheses sin open parentheses beta close parentheses$
=> $s i n squared space alpha plus c o s space left parenthesis alpha plus beta right parenthesis left parenthesis cos open parentheses alpha close parentheses cos open parentheses beta close parentheses minus sin open parentheses alpha close parentheses sin open parentheses beta close parentheses plus 2 s i n space alpha s i n space beta right parenthesis$
=> $s i n squared space alpha plus c o s space left parenthesis alpha plus beta right parenthesis left parenthesis cos open parentheses alpha close parentheses cos open parentheses beta close parentheses plus s i n space alpha s i n space beta right parenthesis$
Step3: Again using the formula of $cos open parentheses alpha plus beta close parentheses equals cos open parentheses alpha close parentheses cos open parentheses beta close parentheses minus sin open parentheses alpha close parentheses sin open parentheses beta close parentheses$ and then multiplying the terms.
$s i n squared space alpha plus left parenthesis cos open parentheses alpha close parentheses cos open parentheses beta close parentheses minus sin open parentheses alpha close parentheses sin open parentheses beta close parentheses right parenthesis left parenthesis cos open parentheses alpha close parentheses cos open parentheses beta close parentheses plus s i n space alpha s i n space beta right parenthesis$
=> $s i n squared space alpha plus left parenthesis cos open parentheses alpha close parentheses squared cos open parentheses beta close parentheses squared minus sin open parentheses alpha close parentheses sin open parentheses beta close parentheses left parenthesis cos open parentheses alpha close parentheses cos open parentheses beta close parentheses plus s i n space alpha s i n space beta left parenthesis cos open parentheses alpha close parentheses cos open parentheses beta close parentheses space minus sin open parentheses alpha close parentheses squared sin open parentheses beta close parentheses squared right parenthesis$
=> $s i n squared space alpha minus sin open parentheses alpha close parentheses squared sin open parentheses beta close parentheses squared plus plus cos open parentheses alpha close parentheses squared cos open parentheses beta close parentheses squared$
Step4: By using the formula $cos open parentheses theta close parentheses squared equals 1 minus sin open parentheses theta close parentheses squared$
=> $sin open parentheses alpha close parentheses squared left parenthesis 1 minus negative sin open parentheses beta close parentheses squared right parenthesis plus plus cos open parentheses alpha close parentheses squared cos open parentheses beta close parentheses squared$
=> $sin open parentheses alpha close parentheses squared cos open parentheses beta close parentheses squared plus plus cos open parentheses alpha close parentheses squared cos open parentheses beta close parentheses squared$
By taking similar terms common we get
$cos open parentheses beta close parentheses squared left parenthesis sin open parentheses alpha close parentheses squared plus plus cos open parentheses alpha close parentheses squared right parenthesis$
Step5: By using the identity $sin open parentheses theta close parentheses squared plus cos open parentheses theta close parentheses squared space equals 1$ we get,
=> $cos open parentheses beta close parentheses squared$

Related Questions to study

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parallel

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parallel

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