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Question

If $alpha comma beta comma gamma text are acute angles and end text cos text end text theta equals sin space beta divided by sin space alpha comma cos space ϕ equals sin space gamma divided by sin invisible function application alpha$ and $cos space left parenthesis theta minus ϕ right parenthesis equals sin space beta sin space gamma$ then the value of $tan squared space alpha minus tan squared space beta minus tan squared space gamma$ is equal to

-1
0
1
2

hint Hint:

In this question, we have to find the value of $tan squared space alpha minus tan squared space beta minus tan squared space gamma$ . For that first we will use the given equation to find the value of one of the three term, then we can put the value obtained in $tan squared space alpha minus tan squared space beta minus tan squared space gamma$ to find the required answer.

The correct answer is: 0

$cos space left parenthesis theta minus ϕ right parenthesis equals sin space beta sin space gamma rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space theta. sin space ϕ close parentheses equals sin space beta. sin space gamma rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space beta sin space gamma close parentheses equals sin space theta. sin space ϕ S q u a r i n g space b o t h space s i d e s comma space w e space g e t colon rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space beta sin space gamma close parentheses squared equals sin squared theta sin squared ϕ rightwards double arrow open parentheses fraction numerator sin beta over denominator sin alpha end fraction fraction numerator sin gamma over denominator sin alpha end fraction minus sin beta. sin gamma close parentheses squared equals open parentheses 1 minus cos squared theta close parentheses open parentheses 1 minus cos squared ϕ close parentheses rightwards double arrow fraction numerator sin squared beta sin squared gamma open parentheses 1 minus sin squared alpha close parentheses over denominator sin squared alpha end fraction equals open parentheses 1 minus fraction numerator sin squared beta over denominator sin squared alpha end fraction close parentheses open parentheses 1 minus fraction numerator sin squared gamma over denominator sin squared alpha end fraction close parentheses rightwards double arrow sin squared beta sin squared gamma open parentheses 1 plus sin to the power of 4 alpha minus 2 sin squared alpha close parentheses equals open parentheses sin squared alpha minus sin squared beta close parentheses open parentheses sin squared alpha minus sin squared gamma close parentheses rightwards double arrow sin squared beta sin squared gamma plus sin squared beta sin squared gamma sin to the power of 4 alpha minus 2 sin squared alpha sin squared beta sin squared gamma equals sin to the power of 4 alpha minus sin squared alpha sin squared beta minus sin squared alpha sin squared gamma plus sin squared beta sin squared gamma rightwards double arrow sin to the power of 4 alpha open parentheses 1 minus sin squared beta sin squared gamma close parentheses minus sin squared gamma open parentheses sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma close parentheses equals 0 rightwards double arrow sin to the power of 4 alpha open parentheses 1 minus sin squared beta sin squared gamma close parentheses equals sin squared gamma open parentheses sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma close parentheses rightwards double arrow sin squared alpha equals fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction...... e q u a t i o n space 1 rightwards double arrow cos squared alpha equals 1 minus fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction rightwards double arrow cos squared alpha equals fraction numerator 1 minus sin squared beta minus sin squared gamma plus sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction..... e q u a t i o n space 2 f r o m space 1 space a n d space 2 tan squared alpha equals fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta minus sin squared gamma plus s in squared beta sin squared gamma end fraction equals fraction numerator sin squared beta minus sin squared beta sin squared gamma plus sin squared gamma minus sin squared beta sin squared gamma over denominator open parentheses 1 minus sin squared beta close parentheses minus sin squared gamma open parentheses 1 minus sin squared beta close parentheses end fraction equals fraction numerator sin squared beta open parentheses 1 minus sin squared gamma close parentheses plus sin squared gamma open parentheses 1 minus sin squared beta close parentheses over denominator open parentheses 1 minus sin squared beta close parentheses open parentheses 1 minus sin squared gamma close parentheses end fraction equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction N o w comma space tan squared alpha minus tan squared beta minus tan squared gamma equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction minus fraction numerator sin squared beta over denominator cos squared beta end fraction minus fraction numerator sin squared gamma over denominator cos squared gamma end fraction equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta minus sin squared beta cos squared gamma minus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction equals 0$

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parallel

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