Question

Use trigonometric identities to verify the expression (1-tan²(x)/sec²(x)) is equal to cos(2x).

Answer 1

To verify the expression (1 - tan²(x)/sec²(x)) is equal to cos(2x), we can use trigonometric identities. By simplifying the expression step-by-step and using identities such as tan²(x) + 1 = sec²(x) and cos²(x) + sin²(x) = 1, we can show that the given expression is equivalent to cos(2x).

Step-by-step explanation:

To verify the expression (1 - tan²(x)/sec²(x)) is equal to cos(2x), we can use trigonometric identities.

1. Start with the expression (1 - tan²(x)/sec²(x))
2. Use the identity tan²(x) + 1 = sec²(x) to rewrite the expression as (1 - tan²(x)/(tan²(x) + 1))
3. Simplify further by multiplying the numerator and denominator by (tan²(x) + 1), which gives ((tan²(x) + 1) - tan²(x))/(tan²(x) + 1)
4. Combine like terms in the numerator to get (1)/(tan²(x) + 1)
5. Use the identity cos²(x) + sin²(x) = 1 to rewrite the expression as (cos²(x))/(cos²(x) + sin²(x))
6. Simplify further by substituting sin²(x) with 1 - cos²(x), which gives (cos²(x))/((cos²(x) + (1 - cos²(x))))
7. Simplify the denominator to get (cos²(x))/1, which equals cos²(x)
8. Rewrite cos²(x) as (1/2)(1 + cos(2x)) using the identity cos(2x) = cos²(x) - sin²(x)
9. Finally, simplify the expression to cos(2x)