Perform the indicated operation and simplify
tan^2(theta)-3sin(theta)tan(theta)sec(theta)
Substitute
sec(θ)= 1/cos(θ)
sin(θ)/cos(θ) = tan(θ)
and simplify.
To perform the operation and simplify the given expression, we need to use trigonometric identities.
First, we'll rewrite the expression using the identities:
tan^2(theta) = sec^2(theta) - 1
tan(theta) = sin(theta)/cos(theta)
sec(theta) = 1/cos(theta)
Now, let's substitute these identities into the given expression:
(sec^2(theta) - 1) - 3sin(theta)(sin(theta)/cos(theta))(1/cos(theta))
Next, let's simplify step by step:
sec^2(theta) - 1 simplifies to tan^2(theta)
sin(theta)(sin(theta)) simplifies to sin^2(theta)
1/cos(theta)(cos(theta)) simplifies to 1
The expression becomes:
tan^2(theta) - 3sin^2(theta)
Finally, let's simplify further if possible:
Use the Pythagorean Identity for sine:
sin^2(theta) + cos^2(theta) = 1
Rearranging the formula:
sin^2(theta) = 1 - cos^2(theta)
Substituting sin^2(theta) into the expression:
tan^2(theta) - 3(1 - cos^2(theta))
Expanding the expression:
tan^2(theta) - 3 + 3cos^2(theta)
The simplified expression is:
tan^2(theta) + 3cos^2(theta) - 3