AP EAMCET Engineering 23 Apr 2019 Shift 1 Paper

Consider the expression,
sin‌x+sin‌y=

√3+1

2

This implies,
2‌sin‌

x+y

2

‌cos‌

x−y

2

=

√3+1

2

x
sin‌

x+y

2

‌cos‌

x−y

2

=

√3+1

4

......(I)
And,
cos‌x+cos‌y=

√3−1

2

Then,
2‌cos‌

x+y

2

‌cos‌

x−y

2

=

√3−1

2

cos‌

x+y

2

‌cos‌ ‌

x−y

2

=

√3−1

4

.....(II)
From equation (I) and (II),
tan‌

x+y

2

=

√3+1

√3−1

Then,
tan2

x+y

2

=

3+1+2√3

3+1−2√3

=

2+√3

2−√3

=

4+3+4√3

4−3

=7+4√3 ......(III)
Square and add equation (III) and (I),
cos2

x−y

2

=

1

16×2(3+1)

=

1

2

sec2

x−y

2

=2
This implies,
tan2

x−y

2

=sec2

x−y

2

−1
= 1
Then,
tan2

x−y

2

+tan2

x+y

2

=1+(7+4√3)
=8+4√3