AP EAMCET Engineering 26 Apr 2017 Shift 1 Paper

Consider the trigonometric equations.
1+cos‌x.cos‌5‌x=sin2x
It is simplified as,
1−sin2x+cos‌x.cos‌5‌x=0
cos2x+cos‌x.cos‌5‌x=0
cos‌x(cos‌x+cos‌5‌x)=0
cos‌x(2‌cos‌3‌x‌cos‌2‌x)=0
Therefore,
cos‌x=0
x=

π

2

,

3π

2

Also,
cos‌3‌x=0
3x=

π

2

,

3π

2

,

5π

2

,

7π

2

,

9π

2

,

11π

2

And,
cos‌2‌x=0
2x=

π

2

,

3π

2

,

5π

2

,

7π

2

Thus, the solution are,
x=

π

2

,

3π

2

,

π

6

,

5π

6

,

7π

6

,

11π

6

,

π

4

,

3π

4

,

5π

4

,

7π

4

Hence, there are 10 solutions.