Trigonometric Identities (HELP)

Question

Use the Reciprocal and Quotient Identities for 1&2

1: Find cscX if sinX= 2√5/5

2: Find cosA if tanA= √2/2 and sinA= -√3/3

Use the Pythagorean Theorem identities for 3&4

3: Find sinX if cotX= -√3/2 and cosX < 0

4: Find cotθ if cosθ= 1/7 and sinθ <  0

Use the cofunction and even-odd identities for 5&6

5: Find tan(-x) if cot(π/2 -x)= -1.84

6: Find cos(θ-π/2) if sinθ= -0.57

Simplify the Trigonometric expressions

7: secX ⋅ sin^2X + cosX

8:(sinθ + 1)(tanθ - secθ)

Prove the Trigonometric expression

1+sec^2X / 1+tan^2X= 1+cos^2X

Answer 1

1. sinx = 2sqr5/5 = 2/sqr5, cscx = sqr5/2
2. tanA = sqr2/2, sinA =-sqr3/3, tanA = sinA/cosA, cosA = sinA/tanA = (-sqr3/3)/(sqr2/2) = (-sqr3/3)(2/sqr2)= -2sqr3/3sqr2 =(-2/3)sqr6/2 = -sqr6/3= CosA
3. cotx = -sqr3/2, cosx <0, sinx is in quadrant II or III, cotx = a/o= adjacent side/opposite side, sinx = opposite side/hypotenuse = o/h, h^2=o^2+a^2 = 3+4 = 7, h=sqr7, sinx =2/sqr7
4. cosT = 1/7= a/h, sinT<0, T is in quadrant IV, cotT = a/o =1/o, o^2 = h^2 -a^2 = 49-1= 48, o=-sqr48, cotT=a/o = -1/sqr48=-sqr48/48=-4sqr3/48=-sqr3/12
5. cot(90-x) = -1.84, tan(-x) =1.84 Period of tangents and cotangents = pi= 180 degrees. tanx and cotx are odd functions. tan(-x) = -tanx,
6. sinT =-.57, cos(T-90) = sinT = -.57, shift the graph of cosT to the right by 90 degrees and the graph of sinT. cosT is an even function cosT=-cos(-T). sinT is an odd function. sin(-T) = -sinT
7. secxsin2x +cosx = (1/cosx)2sinxcosx +cosx = 2sinx +cosx
8. (sinT +1)(tanT-secT) = sinTtanT +tanT -sinTsecT -secT = sinTsinT/cosT +sinT/cosT - sinT/cosT - 1/cosT = (1/cosT)(sin^2(T) -1) = (1/cosT)cos^2(T) = cosT

Prove

(1+sec2x)/(1+tan2x) = 1+cos2x

let x=30, then cos60=1/2, tan60=sqr3/2, sec60=2

(1+sec60)/(1+tan60)= 1+cos60

(1+2)/(1+sqr3/2)=1+1/2

3 +3sqr3/2 = 1 +1/2= 3/2

1+sqr3/2 =1/2

sqr3/2 = -1/2 which is not true,

one counterexample disproves an "identity"

there is no proof of the "identity" IT's not an identity

identities are true for all values of the variable

doesn't prove it, but a clue it may be an identity

1+sec2x = (1+cos2x)(1+tan2x)

use FOIL

1+sec2x = 1 +tan2x +cos2x + tan2xcos2x

subtract 1 from both sides, convert sec2x and tan2x to 1/cos2x and sin2x/cos2x

1/cos2x = cos2x + sin2x/cos2x + (sin2x/cos2x)cos2x

1/cos2x = cos2x + sin2x/cos2x + sin2x

multiply both sides by cos2x

1 = (cos2x)x^2 + sin2x + (sin2x)^2

1 = 1 +sin2x

0 = sin2x

2x = 0 or pi

x = 0 or pi/2or 90 degrees

x= 360n or 90 +360n where n= any integer