tan^2(5x)cos^4(5x) = 1/8 - 1/8cos(20x)

tan^2(5x)cos^4(5x) = 1/8 - 1/8cos(20x)

since tan = sin/cos, you have
sin^2(5x)cos^2(5x) = 1/8 - 1/8cos(20x)

Now use the double-angle formula on the left, and the half-angle formula on the right to get

1/4 sin^2(10x) = 1/4 sin^2(10x)
QED

To prove the given equation tan^2(5x)cos^4(5x) = 1/8 - 1/8cos(20x), we'll break it down into two sides and simplify each side separately.

Left Side:
tan^2(5x)cos^4(5x)

Using the trigonometric identity: tan^2(x) = sec^2(x) - 1

We can write the left side as:

= sec^2(5x)cos^4(5x) - cos^4(5x)

Now, we'll simplify further:

= sec^2(5x)(cos^2(5x))^2 - cos^4(5x)

Using the identity: sec^2(x) = 1 + tan^2(x)

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

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

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

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

= tan^2(5x)cos^4(5x)

We can see that the left side simplifies to tan^2(5x)cos^4(5x).

Right Side:
1/8 - 1/8cos(20x)

We'll simplify the right side:

= 1/8 - 1/8cos(2 * 10x)

Using the double-angle identity: cos(2x) = 2cos^2(x) - 1

= 1/8 - 1/8(2cos^2(10x) - 1)

= 1/8 - 1/8(2cos^2(10x)) + 1/8

= 1/8(1 - 2cos^2(10x) + 1)

= 1/8(2 - 2cos^2(10x))

= 1/4(1 - cos^2(10x))

= 1/4(sin^2(10x))

Now, let's evaluate the left side and the simplified right side:

Left Side: tan^2(5x)cos^4(5x)

Right Side: 1/4(sin^2(10x))

We can see that the left side is equal to the right side, so the provided equation is proven to be true.

To prove this equation, we will start on both sides of the equation and simplify them separately.

Let's start with the left-hand side (LHS) of the equation:

tan^2(5x)cos^4(5x)

To simplify this expression, we can use the trigonometric identity: tan^2(x) = sec^2(x) - 1

And since sec(x) = 1/cos(x), we can rewrite the expression as:

(sec^2(5x) - 1) cos^4(5x)

Expanding the brackets:

cos^4(5x)sec^2(5x) - cos^4(5x)

Now, let's simplify the first term by using the identity: sec^2(x) = 1 + tan^2(x)

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

Expanding the brackets:

cos^4(5x) + cos^4(5x) tan^2(5x) - cos^4(5x)

The cos^4(5x) terms cancel out:

cos^4(5x) tan^2(5x)

Now, let's simplify the right-hand side (RHS) of the equation:

1/8 - 1/8cos(20x)

To simplify this expression, we can factor out 1/8:

1/8 (1 - cos(20x))

Now, let's rewrite cos(20x) using the double-angle identity: cos(2θ) = 2cos^2(θ) - 1

cos(20x) = 2cos^2(10x) - 1

Substituting back into the expression:

1/8 (1 - (2cos^2(10x) - 1))

Simplifying further:

1/8 (1 - 2cos^2(10x) + 1)

1/8 (2 - 2cos^2(10x))

Finally, we can simplify further and combine like terms:

1/8(2(1 - cos^2(10x)))

Using the identity: 1 - cos^2(θ) = sin^2(θ)

1/8 (2sin^2(10x))

This can be further simplified as:

sin^2(10x)/4

Now, comparing the simplified LHS and RHS equations, we see that they are equal:

cos^4(5x) tan^2(5x) = sin^2(10x)/4

Therefore, we have proven the given trigonometric equation.