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Maths-

General

Easy

Question

If $tan space left parenthesis pi cos space theta right parenthesis equals cot space left parenthesis pi sin space theta right parenthesis comma 0 less than theta less than fraction numerator 3 pi over denominator 4 end fraction$ then $sin space open parentheses theta plus pi over 4 close parentheses equals$

$\frac{1}{\sqrt{2}}$
$- \frac{1}{\sqrt{2}}$
$\frac{1}{2 \sqrt{2}}$
$- \frac{1}{2 \sqrt{2}}$

Hint:

In this question, we have to find value of $sin space open parentheses theta plus pi over 4 close parentheses$ if $tan space left parenthesis pi cos space theta right parenthesis equals cot space left parenthesis pi sin space theta right parenthesis comma 0 less than theta less than fraction numerator 3 pi over denominator 4 end fraction$ . We will use the given equation and simplify it further to get the value of the required term.

The correct answer is: $fraction numerator 1 over denominator 2 square root of 2 end fraction$

$space space space space tan left parenthesis pi space cos space ϑ right parenthesis equals c o t space left parenthesis pi space sin space ϑ right parenthesis rightwards double arrow tan left parenthesis pi space cos space ϑ right parenthesis equals tan space open parentheses pi over 2 minus pi space sin space ϑ close parentheses rightwards double arrow pi space cos space ϑ equals pi over 2 minus pi space sin space ϑ rightwards double arrow cos space ϑ equals 1 half minus sin space ϑ rightwards double arrow cos space ϑ plus sin space ϑ equals 1 half M u l t i p l y i n g space b o t h space t h e space s i d e s space w i t h space fraction numerator 1 over denominator square root of 2 end fraction comma space w e space g e t colon rightwards double arrow fraction numerator 1 over denominator square root of 2 end fraction cos space ϑ plus fraction numerator 1 over denominator square root of 2 end fraction sin space ϑ equals fraction numerator 1 over denominator 2 square root of 2 end fraction rightwards double arrow sin space pi over 4 cos space ϑ plus cos space pi over 4 sin space ϑ equals fraction numerator 1 over denominator 2 square root of 2 end fraction rightwards double arrow sin open parentheses ϑ plus pi over 4 close parentheses equals fraction numerator 1 over denominator 2 square root of 2 end fraction$

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