Question

3 tan25°.tan40°.tan50°.tan65° - 1/2tan^2 60°/
4(cos^2 29°+cos^2 61°)​

Answer 1

Note : (1) tan ( 90° - α ) = cot α

. . . . .(2) tan α · cot α = 1

. . . . .(3) cos ( 90° - α ) = sin

Numerator

= 3 tan 25° tan 40° tan 50° tan 65° - (1/2) tan² 60°

= 3 ( tan 25 tan 65 ) ( tan 40 tan 50 ) - (1/2) ( √3 )²

= 3 ( tan 25 cot 25 ) ( tan 40 cot 40 ) - (3/2)

= 3(1)(1) - (3/2)

= 3/2 - (1)

Denominator

= 4 ( cos² 29° + cos² 61° )

= 4 ( sin² 61° + cos² 61° )

= 4(1)

= 4 - (2)

From (1) and (2),

Required Expression = ( 3/2 ) / 4 = 3/8 .