Hi! Can someone help me with these? I'm suppose to verify each idenity. Thanks!
1.) 1/tanθ +tanθ=sec^2θ/tanθ
2.) secθ-1/1-cosθ=secθ
1.
LS = 1/tanØ + tanØ
= (1+tan^2 Ø)/tanØ
but 1+tan^2 Ø = sec^2 Ø, one of the basic identities in your repertoire
= sec^2 Ø/tanØ
= RS
2. for this one you must realize that secØcosØ = 1
LS = (secØ - 1)/(1 - cosØ) , those brackets are essential
= (secØ - 1)/(1 - cosØ) * (1+cosØ)/(1+cosØ)
= (secØ + secØcosØ - 1 - cosØ)/(1 - cos^2 Ø)
= (1/cosØ - cosØ)/sin^2 Ø
= ( (1 - cos^2 Ø)/cosØ )/(sin^2 Ø
= ( sin^2 Ø/cosØ)/sin^2 Ø
= 1/cosØ
= secØ
= RS
Certainly! I'd be happy to help you verify these identities.
For the first identity:
1/tanθ + tanθ = sec^2θ/tanθ.
To verify this identity, we need to simplify both sides of the equation and show that they are equal.
Starting with the left side of the equation, we have:
1/tanθ + tanθ.
To simplify the left side, we can find a common denominator for the two terms. The common denominator is tanθ:
(1 + tan^2θ) / tanθ.
Now, let's simplify the right side of the equation:
sec^2θ/tanθ.
Using the definition of secant, which is 1/cosθ, we can write this as:
(1/cos^2θ) / tanθ.
To simplify further, we can rewrite the expression as:
(1/tanθ) / cos^2θ.
Now, let's simplify both sides of the equation and show that they are equal:
(1 + tan^2θ) / tanθ = (1/tanθ) / cos^2θ.
To do this, we can cross-multiply:
(1 + tan^2θ) * cos^2θ = tanθ.
Expanding the left side of the equation gives us:
cos^2θ + cos^2θ * tan^2θ = tanθ.
Now, using the identity sin^2θ + cos^2θ = 1, we can rewrite the left side of the equation:
(1 - sin^2θ) + cos^2θ * tan^2θ = tanθ.
Simplifying further, we get:
1 - sin^2θ + cos^2θ * tan^2θ = tanθ.
Now, using the Pythagorean identity sin^2θ = 1 - cos^2θ, we can substitute this into the equation:
1 - (1 - cos^2θ) + cos^2θ * tan^2θ = tanθ.
Simplifying the equation gives us:
1 - 1 + cos^2θ + cos^2θ * tan^2θ = tanθ.
Combining like terms, we have:
2cos^2θ + cos^2θ * tan^2θ = tanθ.
Factoring out cos^2θ from the left side, we get:
cos^2θ * (2 + tan^2θ) = tanθ.
Now, dividing both sides by cos^2θ, we have:
2 + tan^2θ = tanθ / cos^2θ.
Using the definition of secant, which is 1/cosθ, we can rewrite the equation:
2 + tan^2θ = tanθ * sec^2θ.
Now, using the identity 1 + tan^2θ = sec^2θ, we can substitute this into the equation:
2 + sec^2θ = tanθ * sec^2θ.
Simplifying further, we get:
2sec^2θ = tanθ * sec^2θ.
Now, dividing both sides by sec^2θ, we have:
2 = tanθ.
Applying the inverse tangent function to both sides, we get:
θ = arctan(2).
Therefore, the identity is not true for all values of θ. The equation is only true when θ is equal to arctan(2).
Let's move on to the second identity.