Verify the identity. Justify each step. tan (thetA) + cot (thetA) = 1 / sin (thetA) cos (thetA)
To prove the given identity, we'll start with the left-hand side (LHS) and work towards the right-hand side (RHS).
LHS: tan(theta) + cot(theta)
Step 1: Rewrite tan and cot in terms of sin and cos:
LHS = sin(theta) / cos(theta) + cos(theta) / sin(theta)
Step 2: Find the common denominator:
LHS = (sin^2(theta) + cos^2(theta)) / (sin(theta) * cos(theta))
Step 3: Apply the Pythagorean identity sin^2(theta) + cos^2(theta) = 1:
LHS = 1 / (sin(theta) * cos(theta))
Step 4: Take the reciprocal of the denominator:
LHS = 1 / sin(theta) * 1 / cos(theta)
RHS: 1 / (sin(theta) * cos(theta))
Since the LHS and RHS are equal, we have verified the identity.
To verify the given trigonometric identity:
tan(θ) + cot(θ) = 1 / (sin(θ) * cos(θ))
We will work on the left-hand side (LHS) of the equation and simplify it until it matches the right-hand side (RHS).
Step 1: Write each trigonometric function in terms of sine and cosine.
tan(θ) = sin(θ) / cos(θ)
cot(θ) = cos(θ) / sin(θ)
Step 2: Substitute the values of tan(θ) and cot(θ) in the original equation.
LHS = sin(θ) / cos(θ) + cos(θ) / sin(θ)
Step 3: Find a common denominator for the fractions.
To find a common denominator, we multiply the first fraction by sin(θ) / sin(θ) and the second fraction by cos(θ) / cos(θ).
LHS = (sin^2(θ) + cos^2(θ)) / (cos(θ) * sin(θ))
Step 4: Apply the Pythagorean Identity sin^2(θ) + cos^2(θ) = 1.
LHS = 1 / (cos(θ) * sin(θ))
Step 5: Rearrange the denominator.
LHS = 1 / (sin(θ) * cos(θ))
Now, the LHS matches the RHS of the equation, concluding that the given identity is verified.
Just remember that
tanx = sinx/cosx
On the LS, replace both tangents , add them up using a common denominator
and recall that sin^2 x + cos^2 x = 1
falls apart so nicely