SOLUTION: a) prove that (1+tan A + sec A)(1+cot A-cosecA)=2 b) Prove that Sin 20degree.sin40degree.sin 60degree.Sin80degree = 3/16 c) sin 5A+2sin7A+sin9A/Cos3A+2cos5A+Cos7A = Sin2A+Cos2A.T

Question 735516: a) prove that (1+tan A + sec A)(1+cot A-cosecA)=2
b) Prove that Sin 20degree.sin40degree.sin 60degree.Sin80degree = 3/16
c) sin 5A+2sin7A+sin9A/Cos3A+2cos5A+Cos7A = Sin2A+Cos2A.Tan5A
d) Show that Sin112+Cos12 / Cos12-Sin12 = Cot35
e) Prove that Sin 7x+Sinx / Cos5x-Cos3x = Sin2x-Cos2x.cotx

Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
a) prove that (1+tan A + sec A)(1+cot A-cosecA)=2
LHS =>

= 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
= sin�60�cos�20 - sin�20�cos�60

cos�A = 1 - sin�A
sin�60�cos�20 - sin�20�cos�60
= sin�60�(1-sin�20) - sin�20�(1-sin�60)
= sin�60- sin�20�sin�60 - sin�20 + sin�20�sin�60
= sin�60 - sin�20
= 3/4 - sin�20

We get
= sin20�[√(3)/2]�(3/4 - sin�20)
= [√(3)/8]�(3sin20 - 4sin�20)

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

(3sin20 - 4sin�20) = sin60

we have
= [√(3)/8]�sin60
= [√(3)/8]�[√(3)/2)
= 3/16
LHS = RHS

RELATED QUESTIONS
1. Simplify a) 1/sin^2x - 1/tan^2x b)sinxcosx/1-sin^2x 2. Prove that tanx + 1/tanx = (answered by nyc_function)

Prove (1 - sin A)/(1 + sin A) = (sec A - tan... (answered by math_tutor2020)

Prove that sin A +cos A-1/sin A -cos A + 1=1/sec A + tan... (answered by Alan3354)

Show that (a) (sec^4-1)/(tan^2x) =2tan^2x (b) (cosec^4x -1)/(cot^2x) =2+cot^2x (c)... (answered by Edwin McCravy,MathTherapy)

If 2(A+B+C)=A Prove that Sin^2 A + Sin^2 B + Sin^2 C=... (answered by Edwin McCravy)

Please help me solve this question: Prove the following identities a. sin {θ}... (answered by tommyt3rd)

prove that tan (A+30�) + cot (A-30�) = 1/sin⁡〖2A -... (answered by advanced_Learner)

Prove that: 1/(cosec A - cot A) - 1/sin A = 1/sin A - 1/(cosec A + cot... (answered by Edwin McCravy)

1.Derive a formula for cos 2A with aid of cos (A + B). 2.Prove that 2 tan x + 2 cot x =... (answered by Edwin McCravy)