Expert: Socrates - 12/2/2009

Question
i am having a problem solving these trigs. please could you help for my exam 1. prove that tanA+ cotA =2cosecA.SecA
2.COS 70+ COS 30 =Cos 50 3.  evaluate Tan (345)without using calculator
4. if p= cosx+sinx and q =COSX-SINX, PROVE THAT 1.pq=1-2Sin2 x
p2+q2=2 thank you.

Answer
1)


tanA + cot A =

sinA/cosA + cosA/sinA =

(common denominator is sinAcosA)

(sinA)^2/sinAcosA + (cosA)^2/sinAcosA =

( (sinA)^2 + (cosA)^2 )/sinAcosA =

1/sinAcosA =

(1/sinA) (1/cosA) =

cscA secA



2)


This is not true. Did you copy the problem incorrectly? You can check that this is not true by using your calculator



3)


Tan 345 = Tan ( 120 + 225 )

now use the formula for the tangent of a sum  Tan (A+B) = (TanA + TanB)/ (1-TanATanB)



Tan 120 = -√3

Tan 225 = 1


so

Tan ( 120 + 225 ) = (Tan120 + Tan225)/ (1-Tan120Tan225)


= (-√3 + 1 )/ (1-(-√3)(1)) =

(-√3 + 1 )/ (1 + √3))

you can rationalize the denominator by multiplying numerator and denominator by 1 - √3

= √3 - 2


4)


p = cosx + sinx

q = cosx - sinx


pq = (cosx + sinx)(cosx - sinx) = (cosx)^2 - (sinx)^2  =  (1 - (sinx)^2 ) - (sinx)^2 =

1 - 2(sinx)^2


p^2 + q^2 = (cosx + sinx)^2 +  (cosx - sinx)^2 =

(cosx)^2 + 2sinxcosx + (sinx)^2 + (cosx)^2 - 2sinxcosx + (sinx)^2 =

2(cosx)^2 + 2(sinx)^2 = 2((cosx)^2 + (sinx)^2 ) = 2