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Given: sin ∅= 4/5 and cos x = -5/13 ; evaluate the following expression.

tan( ∅ - x )

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- cosø=3/5
- sinx=12/13

Now

cos(ø-x)

- cosøcosx+sinøsinx
- (3/5)(-5/13)+(4/5)(12/13)
- (33/65)

sin(ø-x)

- sinøsinx-cosøcosx
- 48/65+33/65
- 81/65

So

tan(ø-x)

- sin(ø-x)/cos(ø-x)
- 81/65÷33/65
- 81/33
- 27/11
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By definition of tangent,

tan(θ - x) = sin(θ - x) / cos(θ - x)

Expand the sine and cosine terms using the angle sum identities,

sin(x ± y) = sin(x) cos(y) ± cos(x) sin(y)

cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y)

from which we get

tan(θ - x) = (sin(θ) cos(x) - cos(θ) sin(x)) / (cos(θ) cos(x) + sin(θ) sin(x))

Also recall the Pythagorean identity,

cos²(x) + sin²(x) = 1

from which we have two possible values for each of cos(θ) and sin(x):

cos(θ) = ± √(1 - sin²(θ)) = ± 3/5

sin(x) = ± √(1 - cos²(x)) = ± 12/13

Since there are 2 choices each for cos(θ) and sin(x), we'll have 4 possible values of tan(θ - x) :

• cos(θ) = 3/5, sin(x) = 12/13 :

tan(θ - x) = -56/33

• cos(θ) = -3/5, sin(x) = 12/13 :

tan(θ - x) = 16/63

• cos(θ) = 3/5, sin(x) = -12/13 :

tan(θ - x) = -16/63

• cos(θ) = -3/5, sin(x) = -12/13 :

tan(θ - x) = 56/33
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