Question 735516
a) prove that (1+tan A + sec A)(1+cot A-cosecA)=2

LHS =>{{{ (1 + (sin A/cos A)+(1/cos A))(1+(cos A/sin A)-(1/sin A))}}}

{{{ (( cos A + sin A +1)/cos A)((sin A +cos A -1)/sin A)}}}



{{{ ((cos A +sin A )^2 -1)/sin A cos A}}}


{{{( cos^2 (A )+sin ^2( A) +2*sin A cos A -1)/(sin A cos A)}}}




{{{(1+2 sin A cosA-1)/sin A cos A}}}


= 2

LHS = RHS



b) Prove that Sin 20degree.sin40degree.sin 60degree.Sin80degree = 3/16


 sin(A+B) = sin A cos B + cos A sin B


 
sin(40) as sin(60-20)
sin(80) as sin(60+20)
 
 sin(20)•sin(60-20)•[√(3)/2]•sin(60+20)

- Simplify sin(60-20)•sin(60+20)
 sin(60-20) = sin60•cos20 - sin20•cos60
 sin(60+20)= sin60•cos20 + sin20•cos60
 multiply 
 = sin²60•cos²20 - sin²20•cos²60


cos²A = 1 - sin²A

sin²60•cos²20 - sin²20•cos²60
 = sin²60•(1-sin²20) - sin²20•(1-sin²60)
 = sin²60- sin²20•sin²60 - sin²20 + sin²20•sin²60
 = sin²60 - sin²20
 = 3/4 - sin²20

 
 We get
 = sin20•[√(3)/2]•(3/4 - sin²20)
 = [√(3)/8]•(3sin20 - 4sin³20)

 
 we know that
 sin(3A) = 3sin(A) - 4sin³(A)
 

(3sin20 - 4sin³20) = sin60

 
we have
 = [√(3)/8]•sin60
 = [√(3)/8]•[√(3)/2)
 = 3/16

LHS = RHS