Sarah asked in Science & MathematicsMathematics · 8 years ago

# HELP SOLVING THESE TRIG. IDENTITIES PLEASE I BEG OF YOU BBY?

Make sure to show steps! Explain if possible!

9.) cos(x+y)cosy+sin(x+y)siny=cosx

10.) sin x- tan y cos x= (sin(x-y)/cos y)

12.) (tan(π/4 + x) - tan (π/4-x))/(tan(π/4+x) +tan (π/4-x)) = 2sinxcosx

13.) sin (x+y)sin(x=y)=cos²y-cos²x

14.) tan (x+y) tan (x-y)= (sin²x-sin²y)/(cos²x-sin²y)

Update:

@michaelempeigne I meant (x-y) sorry mate, typo :(

Relevance
• 8 years ago

9) LHS = cos ( x + y - y )

LHS = cos x

LHS = RHS

QED

10) LHS = sin x - ( sin y / cos y )*cos x

LHS = (sin x cos y - sin y cos x ) / cos y

LHS = sin ( x - y ) / cos y

LHS = RHS

QED

12) LHS = { [ sin ( pi / 4 + x ) cos (pi / 4 - x ) - sin ( pi / 4 - x ) cos ( pi /4 + x ) ] / [ cos ( pi / 4 - x ) cos ( pi / 4 + x ) ] } / { [ sin ( pi / 4 + x ) cos ( pi / 4 - x ) + sin ( pi / 4 - x ) cos ( pi / 4 + x ) ] / [ cos ( pi / 4 - x ) cos ( pi / 4 + x ) ] }

LHS = sin ( pi / 4 + x - pi / 4 + x ) / sin ( pi / 4 + x + pi / 4 - x )

LHS = sin 2x / sin ( pi / 2 )

LHS = sin 2x

LHS = 2 sin x cos x

LHS = RHS

QED

13) what do you mean by sin ( x = y ) ??

14) LHS = tan ( x + y ) tan ( x - y )

LHS = [ ( tan x + tan y ) / ( 1 - tan x tan y ) ] [ (tan x - tan y ) / ( 1 + tan x tan y ) ]

LHS = ( tan^2 x - tan^2 y ) ( 1 - tan^2 xtan^2 y )

LHS = [ ( cos^2 y sin^2 x - sin^2 y cos^2 x ) / ( cos^2 y cos^2 x ) ] / [ ( cos^2 x cos^2 y - sin^2 x sin^2 y ) / (cos^2 x cos^2 y ) ]

LHS = (cos^2 y sin^2 x - sin^2 y cos^2 x ) / ( cos^2 x cos^2 y - sin^2 x sin^2 y )

LHS = ( sin^2 x - sin^2 x sin^2 y - sin^2 y + sin^2 x sin^2 y ) / ( cos^2 x - cos^2 x sin^2 y - sin^2 y + cos^2 x sin^2 y )

LHS = ( sin^2 x - sin^2 y ) / ( cos^2 x - sin^2 y )

LHS = RHS

QED

Source(s): my brain
• 8 years ago

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

sin(a + b) = sin(a)cos(b) + sin(b)cos(a)

sin(a - b) = sin(a)cos(b) - sin(b)cos(a)

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b))

cos(x + y)cos(y) + sin(x + y)sin(y) =>

cos(x + y - y) =>

cos(x)

sin(x) - tan(y)cos(x) =>

sin(x) - sin(y)cos(x) / cos(y) =>

(sin(x) * cos(y) - sin(y) * cos(x)) / cos(y) =>

sin(x - y) / cos(y)

tan(pi/4 + x) =>

(tan(pi/4) + tan(x)) / (1 - tan(pi/4) * tan(x)) =>

(1 + tan(x)) / (1 - tan(x))

tan(pi/4 - x) =>

(tan(pi/4) - tan(x)) / (1 + tan(pi/4) * tan(x)) =>

(1 - tan(x)) / (1 + tan(x))

1 - tan(x) = a

1 + tan(x) = b

(tan(pi/4) + x) - tan(pi/4 - x)) / (tan(pi/4 + x) + tan(pi/4 - x)) =>

(b/a - a/b) / (b/a + a/b) =>

((b^2 - a^2) / (ab)) / ((b^2 + a^2) / (ab)) =>

(b^2 - a^2) / (b^2 + a^2) =>

((1 + tan(x))^2 - (1 - tan(x))^2) / ((1 + tan(x))^2 + (1 - tan(x))^2) =>

(1 + 2tan(x) + tan(x)^2 - 1 + 2tan(x) - tan(x)^2) / (1 + 2tan(x) + tan(x)^2 + 1 - 2tan(x) + tan(x)^2) =>

4 * tan(x) / (2 + 2 * tan(x)^2) =>

2 * tan(x) / (1 + tan(x)^2) =>

2 * tan(x) / sec(x)^2 =>

2 * (sin(x)/cos(x)) / (1/cos(x)^2) =>

2 * sin(x) * cos(x)^2 / cos(x) =>

2 * sin(x) * cos(x)

sin(x + y) * sin(x - y) =>

(sin(x)cos(y) + sin(y)cos(x)) * (sin(x)cos(y) - sin(y)cos(x)) =>

sin(x)^2 * cos(y)^2 - sin(y)^2 * cos(x)^2 =>

(1 - cos(x)^2) * cos(y)^2 - (1 - cos(y)^2) * cos(x)^2 =>

cos(y)^2 - cos(x)^2 * cos(y)^2 - cos(x)^2 + cos(x)^2 * cos(y)^2 =>

cos(y)^2 - cos(x)^2

tan(x + y) * tan(x - y) =>

(tan(x) + tan(y)) * (tan(x) - tan(y)) / (((1 - tan(x)tan(y)) * (1 + tan(x)tan(y))) =>

(tan(x)^2 - tan(y)^2) / (1 - tan(x)^2 * tan(y)^2) =>

(sin(x)^2 / cos(x)^2 - sin(y)^2 / cos(y)^2) / (1 - sin(x)^2 * sin(y)^2 / (cos(x)^2 * cos(y)^2)) =>

((sin(x)^2 * cos(y)^2 - sin(y)^2 * cos(x)^2) / (cos(x)^2 * cos(y)^2)) / ((cos(x)^2 * cos(y)^2 - sin(x)^2 * sin(y)^2) / (cos(x)^2 * cos(y)^2)) =>

(sin(x)^2 * cos(y)^2 - sin(y)^2 * cos(x)^2) / (cos(x)^2 * cos(y)^2 - sin(x)^2 * sin(y)^2)) =>

(sin(x)^2 * (1 - sin(y)^2) - sin(y)^2 * (1 - sin(x)^2)) / (cos(x)^2 * (1 - sin(y)^2) - (1 - cos(x)^2) * sin(y)^2)) =>

(sin(x)^2 - sin(x)^2 * sin(y)^2 - sin(y)^2 + sin(x)^2 * sin(y)^2) / (cos(x)^2 - cos(x)^2 * sin(y)^2 - sin(y)^2 + cos(x)^2 * sin(y)^2) =>

(sin(x)^2 - sin(y)^2) / (cos(x)^2 - sin(y)^2)