Question

Solve for Tan^4(5x) using power reduction formulas

Answer 1

Step-by-step explanation:

Solve for Tan^4(5x) using power reduction formulastan^4(5x)

= (tan^2(5x))^2

= (sec^2(5x) - 1)^2 (using the formula tan^2(x) = sec^2(x) - 1)

= sec^4(5x) - 2(sec^2(5x)) + 1

To solve for tan^4(5x) using power reduction formulas, we can start by applying the formula tan^2(x) = sec^2(x) - 1 repeatedly:Now, we can use the formula sec^2(x) = 1 + tan^2(x) to express sec^2(5x) in terms of tan^2(5x):Substituting this into our previous equation, we get:tan^4(5x) = (1 + tan^2(5x))^2 - 2(1 + tan^2(5x)) + 1

= 1 + 2tan^2(5x) + tan^4(5x) - 2 - 2tan^2(5x) + 1

= tan^4(5x) - 1

Therefore, tan^4(5x) = sec^4(5x) - 2(sec^2(5x)) + 1 = (1 + tan^2(5x))^2 - 2(1 + tan^2(5x)) + 1 = tan^4(5x) - 1.

Note that we could have also used the formula tan^2(x) = 1/csc^2(x) - 1 instead of tan^2(x) = sec^2(x) - 1 in the first step to arrive at the same result.

Answer 2

To solve for Tan^4(5x) using power reduction formulas, apply the formula Tan^4(θ) = [Tan^2(θ)]^2 - 2[Sin^2(θ)][Cos^2(θ)]. Substitute θ = 5x and use the identity Sin^2(θ) + Cos^2(θ) = 1 to simplify the equation.

Step-by-step explanation:

In order to solve for Tan^4(5x) using power reduction formulas, we can apply the formula Tan^4(θ) = [Tan^2(θ)]^2 - 2[Sin^2(θ)][Cos^2(θ)]. This formula allows us to reduce higher powers of tangent to lower powers of tangent and trigonometric functions. In this case, let's substitute θ = 5x:

So, Tan^4(5x) = [Tan^2(5x)]^2 - 2[Sin^2(5x)][Cos^2(5x)]

Remember that Sin^2(θ) + Cos^2(θ) = 1. Using this identity, we can further simplify the equation:

Tan^4(5x) = [Tan^2(5x)]^2 - 2[(1 - Cos^2(5x))][Cos^2(5x)]

Tan^4(5x) = [Tan^2(5x)]^2 - 2[1][Cos^2(5x)] + 2[Cos^4(5x)]