need help don't understand how to do this....
1.Verify the identity cot(θ-( π/2)= -tan θ
2. Verify the identity tan θ + cot θ =1/sin θ cosθ
a. Personally, I would change the left side to cos(theta+PI/2) /sin(theta+PI/2)
then do the addition formulas in the numerator, and denominator, and it ought to reduce quickly to -sinTheta/cosTheta
b. Easy. Change the left side to sin/cos+cos/sin, combine to denominator cosThetaSinTheta, it reduces immediately.
Thanks
Sure, I'd be happy to help you understand how to verify these identities.
1. To verify the identity cot(θ - (π/2)) = -tan θ, we need to simplify both sides of the equation and show that they are equal to each other.
Starting with the left-hand side (LHS), we have cot(θ - (π/2)). The cotangent function is defined as 1/tan, so we can rewrite this expression as 1/tan(θ - (π/2)).
Now, by using the angle subtraction formula for tangent, we can rewrite tan(θ - (π/2)) as -tan(θ), because tan(θ - (π/2)) = -tan(θ).
Therefore, the LHS becomes 1/(-tan θ), which is equivalent to -1/tan θ.
Now, let's simplify the right-hand side (RHS), which is -tan θ. By multiplying the numerator and denominator of -tan θ by -1, we get 1/tan θ.
Thus, we have shown that -1/tan θ is equal to 1/tan θ, and therefore, we have verified the identity cot(θ - (π/2)) = -tan θ.
2. To verify the identity tan θ + cot θ = 1/(sin θ cos θ), we can start by expressing cot θ in terms of sine and cosine.
The cotangent function is defined as 1/tan θ. So, cot θ = 1/tan θ = cos θ/sin θ.
Now, let's substitute cot θ with its equivalent expression in the left-hand side (LHS) of the equation:
LHS = tan θ + cot θ
= tan θ + (cos θ / sin θ).
Next, we can find a common denominator for the two terms. The common denominator is sin θ.
So the LHS becomes: (sin θ * tan θ + cos θ) / sin θ.
Now, let's simplify the numerator:
The tangent function is defined as sin θ / cos θ. So, sin θ * tan θ = sin θ * (sin θ / cos θ) = sin^2 θ / cos θ.
Therefore, the LHS becomes: (sin^2 θ / cos θ + cos θ) / sin θ.
To simplify the expression further, we can put the numerator over a common denominator:
LHS = (sin^2 θ + cos^2 θ) / (cos θ * sin θ).
Now we recognize that sin^2 θ + cos^2 θ is equal to 1 (a fundamental identity).
So the LHS simplifies to: 1 / (cos θ * sin θ).
Thus, we have shown that LHS is equal to 1 / (cos θ * sin θ), the same as the right-hand side (RHS).
Therefore, we have verified the identity tan θ + cot θ = 1/(sin θ cos θ).