Prove the trig identity
cos(4x)=2-sec^2(2x)/sec^2(2x)
To prove the trigonometric identity cos(4x) = 2 - sec^2(2x)/sec^2(2x), we can start with the right-hand side (RHS) of the equation and manipulate it until it matches the left-hand side (LHS).
1. Begin with the RHS: 2 - sec^2(2x)/sec^2(2x)
2. Rewrite the RHS in terms of cosine and secant: 2 - 1/cos^2(2x) * 1/cos^2(2x)
3. Combine the fractions: 2 - 1/cos^2(2x * cos^2(2x)
4. Simplify the expression inside the parentheses: 2 - 1/cos^4(2x)
5. Use the Pythagorean identity sec^2(theta) = 1 + tan^2(theta): cos^2(2x) = 1 + tan^2(2x)
6. Substitute the expression from step 5 into the equation: 2 - 1/(1 + tan^2(2x))^2
7. Expand the denominator: 2 - 1/(1 + 2tan^2(2x) + tan^4(2x))
8. Simplify: 2 - 1/(1 + 2tan^2(2x) + tan^4(2x))
9. Use the Pythagorean identity tan^2(theta) = sec^2(theta) - 1: 2 - 1/(1 + 2(sec^2(2x) - 1) + (sec^2(2x) - 1)^2)
10. Expand the expression: 2 - 1/(1 + 2sec^2(2x) - 2 + sec^4(2x) - 2sec^2(2x) + 1)
11. Simplify and combine like terms: 2 - 1/(sec^4(2x) + sec^2(2x))
12. Apply the Pythagorean identity for the secant function: 2 - 1/(sec^2(2x)(sec^2(2x) + 1))
13. The denominator sec^2(2x)(sec^2(2x) + 1) can be simplified to sec^2(2x) + sec^2(2x) = 2sec^2(2x).
14. Substitute the simplified denominator into the equation: 2 - 1/(2sec^2(2x))
15. The resulting expression is equivalent to 2 - sec^2(2x)/sec^2(2x), which matches the LHS.
Therefore, the trigonometric identity cos(4x) = 2 - sec^2(2x)/sec^2(2x) has been proven.