Trigonometric Ratio and Identites

Appeared 45 times across 14 years of JEE Main papers — about 3.2 questions a year. Every one of them is below, with a full worked solution.

Questions per year

2’04

1’06

1’10

1’11

1’13

1’14

1’16

1’17

7’19

4’20

8’21

6’22

5’23

6’24

01Medium45×since 2004Q5633

Let

\alpha ,\,\beta

be such that

\pi < \alpha - \beta < 3\pi

. If

sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}

and

\cos \alpha + \cos \beta = - {{27} \over {65}}

then the value of

\cos {{\alpha - \beta } \over 2}