If sin x + cos x = sqrt{y+(1/y)} such that x belongs to [0,6Π] and y>0 prove that x = Π/4 or 9Π/4.

Asked by acv27joy | 8th Oct, 2018, 09:42: AM

Expert Answer:

sin x plus cos x equals square root of y plus 1 over y end root
l e t space y equals space tan space theta
sin x plus cos x equals square root of tan theta plus fraction numerator 1 over denominator tan theta end fraction end root
sin x plus cos x equals square root of fraction numerator 1 plus tan squared theta over denominator tan theta end fraction end root
sin x plus cos x equals square root of fraction numerator s e c squared theta over denominator tan theta end fraction end root
sin x plus cos x equals fraction numerator s e c space theta over denominator square root of tan theta end root end fraction
sin x plus cos x equals fraction numerator begin display style fraction numerator 1 over denominator cos space theta end fraction end style over denominator square root of begin display style fraction numerator sin theta space over denominator cos theta end fraction end style end root end fraction

sin x plus cos x equals fraction numerator 1 over denominator square root of sin theta space cos theta end root end fraction

s q u a r i n g space o n space b o t h space s i d e
sin squared x plus 2 sin x space cos x plus cos squared x equals fraction numerator 1 over denominator sin theta space cos theta end fraction
1 plus 2 sin x space cos x equals fraction numerator 1 over denominator sin theta space cos theta end fraction
2 sin x space cos x equals fraction numerator 1 over denominator sin theta space cos theta end fraction minus 1
sin 2 x equals fraction numerator 2 over denominator sin 2 theta end fraction minus 1
sin 2 x space sin space 2 theta equals 2 minus sin 2 theta
sin 2 x plus sin 2 x sin 2 theta space equals space 2
n o w space a d d i t i o n space o f space t w o space sin space f u n c t i o n space i s space 2
s o space b o t h space o f space t h e m space s h o u l d space e q u a l space t o space 1 space a s space minus 1 less or equal than sin space x less or equal than 1 space f o r space a l l space x
s o space sin 2 x equals 1
s o space 2 x equals straight pi over 2 space o r space space 2 x equals fraction numerator 5 straight pi over denominator 2 end fraction space o r space space 2 x equals fraction numerator 9 straight pi over denominator 2 end fraction space space a s space x space l i e s space i n left square bracket 0 comma 6 straight pi right square bracket
so space straight x equals straight pi over 4 space space or space space straight x equals fraction numerator 5 straight pi over denominator 4 end fraction space space or space space straight x equals fraction numerator 9 straight pi over denominator 4 end fraction

Answered by Ram Singh | 19th Oct, 2018, 07:32: PM