Verify and identify, I am so confused on how to do this? this is an extra credit but I am lost, please help.
cot(�Ý-ƒÎ/2)=-tan �Ý
I do not understand your symbols, do not have whatever font you are using. I assume this is some kind of a phase shift in the argument of the cotangent.
If you mean
cot(θ-π/2) then that's easy
just as sinθ = cos(π/2-θ),
tanθ = cot(π/2-θ)
and, since cot(-θ) = -cotθ,
cot(θ-π/2) = -cot(π/2-θ) = -tanθ
Oh, well, if you must, show that
cot(θ-π/2) = 1/tan(θ-π/2)
= (1+tanθ*tan π/2)/(tanθ - tanπ/2)
Now divide by tanθ*tan π/2 and you have
= (cotθ*cot π/2+1)/(cotπ/2 - cotθ)
= 1/(0-cotθ)
= -tanθ
To verify and identify the given equation cot(θ - π/2) = -tan(θ), we can use the trigonometric identities for cotangent and tangent.
Here's how you can verify it step by step:
Step 1: Start with the left side of the equation: cot(θ - π/2).
Step 2: Rewrite cotangent as the reciprocal of the tangent function: 1/tan(θ - π/2).
Step 3: Use the angle difference formula for the tangent function: tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B)). In this case, θ - π/2 is the angle difference, and θ is A, while π/2 is B.
Step 4: Plug in the values into the angle difference formula: tan(θ - π/2) = (tan(θ) - tan(π/2))/(1 + tan(θ)tan(π/2)).
Step 5: Simplify tan(π/2) using its identity: tan(π/2) is undefined since it corresponds to a vertical asymptote in the tangent graph.
Step 6: Since tan(π/2) is undefined, the denominator of the equation becomes zero: [(tan(θ) - undefined)/(1 + tan(θ)undefined)].
Step 7: Any division by zero is undefined, so the left side of the equation remains undefined.
Step 8: Now, let's examine the right side of the equation: -tan(θ).
Step 9: From the identity tan(-θ) = -tan(θ), we can rewrite the right side as -tan(θ - 0), which simplifies to -tan(θ).
Step 10: We can see that the right side of the equation is equal to the left side, which is undefined.
Therefore, we can conclude that the equation cot(θ - π/2) = -tan(θ) is true for all values of θ, except when the left side is undefined.