Prove Trig. Identities
1. sec è (sec è - cos è)= tan^2 è
2. tan^2 è (1 + cot^2 è) = sec^2 è
To prove the trigonometric identities, we'll start by manipulating the left side of each equation until it matches the right side.
1. sec(θ) (sec(θ) - cos(θ)) = tan^2(θ)
Starting with the left side,
sec(θ) (sec(θ) - cos(θ))
We can rewrite sec(θ) as 1/cos(θ),
(1/cos(θ)) (sec(θ) - cos(θ))
Expanding the expression,
(sec(θ)/cos(θ)) - (cos(θ)/cos(θ))
Combine the fractions,
(sec(θ) - cos(θ))/cos(θ)
Now, we need to simplify the numerator using the Pythagorean identity,
sec^2(θ) - 1
Substituting back into our expression,
(sec^2(θ) - 1)/cos(θ)
Using the identity tan(θ) = sin(θ)/cos(θ),
(sec^2(θ) - 1)/(1/cos(θ))
Multiply by the reciprocal,
(sec^2(θ) - 1)(cos(θ)/1)
Distribute,
(sec^2(θ)cos(θ) - cos(θ))/1
Divide by cos(θ),
sec^2(θ) - cos(θ)
We have successfully simplified the left side to match the right side, tan^2(θ). Therefore, the identity sec(θ) (sec(θ) - cos(θ)) = tan^2(θ) is proved.
2. tan^2(θ) (1 + cot^2(θ)) = sec^2(θ)
Starting with the left side,
tan^2(θ) (1 + cot^2(θ))
Using the identity cot(θ) = 1/tan(θ),
tan^2(θ) (1 + (1/tan^2(θ)))
Combine the fractions,
tan^2(θ) (1 + tan^2(θ)/1)
Multiply by the reciprocal,
tan^2(θ) (tan^2(θ) + 1)
Distribute,
tan^4(θ) + tan^2(θ)
Using the Pythagorean identity tan^2(θ) + 1 = sec^2(θ),
tan^4(θ) + (sec^2(θ) - 1)
Combine like terms,
tan^4(θ) + sec^2(θ) - 1
We have successfully simplified the left side to match the right side, sec^2(θ). Therefore, the identity tan^2(θ) (1 + cot^2(θ)) = sec^2(θ) is proved.
To prove trigonometric identities, we generally convert one or both sides of the equation to terms of a single trigonometric function or use the fundamental trigonometric identities to simplify the equation. Let's go through the proofs for the given trigonometric identities:
1. sec(θ) * (sec(θ) - cos(θ)) = tan^2(θ)
To prove this identity, we'll only manipulate the left-hand side (LHS) of the equation:
Start with the LHS:
LHS = sec(θ) * (sec(θ) - cos(θ))
Now, let's simplify using the fundamental trigonometric identity:
sec(θ) = 1/cos(θ)
Replace sec(θ) in the equation:
LHS = (1/cos(θ)) * (1/cos(θ) - cos(θ))
To simplify further, we'll find the least common denominator (LCD) between the two terms:
LHS = (1/cos(θ)) * ((1 - cos^2(θ))/cos(θ))
Using the trigonometric identity:
sin^2(θ) + cos^2(θ) = 1
Substitute sin^2(θ) = 1 - cos^2(θ) into the equation:
LHS = (1/cos(θ)) * (sin^2(θ)/cos(θ))
Now, simplify the expression:
LHS = (sin^2(θ))/(cos^2(θ))
Using the definition of tangent:
tan^2(θ) = (sin^2(θ))/(cos^2(θ))
Therefore, LHS = RHS, and we have proved the given trigonometric identity.
2. tan^2(θ) * (1 + cot^2(θ)) = sec^2(θ)
Let's manipulate the left-hand side (LHS) of the equation:
Start with the LHS:
LHS = tan^2(θ) * (1 + cot^2(θ))
Using the reciprocal trigonometric identities:
tan(θ) = sin(θ)/cos(θ) and cot(θ) = cos(θ)/sin(θ)
Replace the trigonometric functions in the equation:
LHS = (sin^2(θ)/cos^2(θ)) * (1 + (cos^2(θ)/sin^2(θ)))
Simplify further:
LHS = (sin^2(θ) + cos^2(θ))/(cos^2(θ)/sin^2(θ))
Using the trigonometric identity:
sin^2(θ) + cos^2(θ) = 1
Simplify the expression:
LHS = 1/(cos^2(θ)/sin^2(θ))
Using the reciprocal identity:
tan^2(θ) = (sin^2(θ)/cos^2(θ)) = (1/cos^2(θ)) / (1/sin^2(θ)) = (1/cos^2(θ)) * (sin^2(θ)/1)
Replace tan^2(θ) in the equation:
LHS = tan^2(θ)/1
Finally, simplifying the expression:
LHS = tan^2(θ)
Therefore, LHS = RHS, and we have proved the given trigonometric identity.