How do I verify that the following equations is an identity: 1+ sec x sin x tan x = sec^2 x
Do I change the sec and tan to sin and cos equivalent?
Yes. I showed you how in your other post of the same question.
To verify whether the given equation is an identity, we need to simplify one side of the equation and see if it equals the other side. In this case, we'll simplify the left side of the equation and check if it is equal to the right side.
Starting with the left side of the equation:
1 + sec(x)sin(x)tan(x)
To simplify this expression, we can use the trigonometric identities:
sec(x) = 1/cos(x)
sin(x) = sin(x)
tan(x) = sin(x)/cos(x)
Let's substitute these values into the equation:
1 + (1/cos(x))sin(x)(sin(x)/cos(x))
Next, we can combine the terms by multiplying:
1 + (1/cos(x))(sin^2(x)/cos(x))
To simplify further, we can multiply both terms by cos(x) to get rid of the fractions:
cos(x) + (sin^2(x)/cos(x))
Now, let's simplify the fraction by dividing the numerator by the denominator:
cos(x) + sin^2(x)/cos(x) = cos(x) + (1 - cos^2(x))/cos(x)
Simplifying the expression inside the parentheses:
cos(x) + 1/cos(x) - cos^2(x)/cos(x) = cos(x) + 1/cos(x) - cos(x)
At this point, we can combine like terms:
1/cos(x) - cos(x) + cos(x) = 1/cos(x)
Notice that we arrived at the right side of the equation, which is sec^2(x).
Therefore, since the left side of the equation simplifies to the right side, we can conclude that the given equation, 1 + sec(x)sin(x)tan(x) = sec^2(x), is indeed an identity.