Question

Sin17 sin3 sin1 sec73 sec87 sec89

Answer 1

Step-by-step explanation:

(sin1°) (sin3°) (sin5°) (sin7°) … (sin89°)=1/2^n=x

Find n

Answer :

n = 44.5=89/2

Solution :

(sin89°) =cos1°

(sin87°) =cos3°

And so on till 47°

x=(sin1°×cos1°)(sin3°×cos3°)(sin5°×cos5°)……sin45°

Multiple and divide every bracket with 2

= (1/2 ^22)(sin45°)(sin2°)(sin6°)(sin10°)……(sin82°)(sin86°)

Sin45°=1/2^.5 (sin@)=(cos90°-@)

=(1/2 ^22.5)(cos4°)(cos8°)(cos12°)……(cos84°)(cos88°)

Let,

p=(cos4°)(cos8°)(cos12°)……(cos84°)(cos88°)

And

q=(sin4°)(sin8°)(sin12°)……(sin84°)(sin88°)

So,

p*q=(sin4°×cos4°)(sin8°×cos8°)(sin12°×cos12°)……. (sin88°×cos88°)

Multiple and divide every bracket with 2

=(1/2^22)(sin8°)(sin16°)(sin24°)…(sin88°)(sin96°)…(sin168°)(sin176°)

(sin@) =(sin180°-@)

=(1/2^22)(sin8°)(sin16°)(sin24°)…(sin88°)(sin84°)…(sin12°)(sin4°)

Thus,

p*q=(1/2^22)*q (q! =0)

So,

p=(1/2^22)

x=(1/2^22. 5)*p

=(1/2^22. 5)(1/2^22)

=(1/2^44.5)