Question

Find the value of tan1° tan 2° tan 3° ..........tan89° sin (90°-20°)cos 20°+cos(90°-20°) sin20° .

Answer 1

Tan1.tan2......tan89
But tan can be written as cot 1 [tan{90-1}=cot1] by tan (90-theta)=cot(theta)
Likewise...........,
Tan88=cot2
Tan87=cot3
......up to tan46=cot44 then middle one is tan45=1
So it is tan1. tan....... Tan44.tan45.cot 44....cot tan and cot cancel out by [tan(theta)*cot(theta)=1]

so ....remaining is 1

So now our equation reduces to sin(90-20)cos20 +cos(90-20)sin20

• sin(90-theta)=cos(theta)
• cos(90-theta)=sin(theta)

therefore cos20cos20+sin20sin20
Cos²20+sin²20 =1.

• sin²(theta)+cos²(theta)=1.

so answer is 1.