Proof:

Let's start by using the given information:

sec(theta) = x^(1/4x)

We know that sec(theta) is the reciprocal of cos(theta), so we can rewrite the equation as:

1/cos(theta) = x^(1/4x)

Step 1: Expressing cos(theta) in terms of sec(theta)

To solve for cos(theta), we can take the reciprocal of both sides of the equation:

cos(theta) = 1 / (x^(1/4x))

Step 2: Expressing tan(theta) in terms of sin(theta) and cos(theta)

Next, let's express tan(theta) in terms of sin(theta) and cos(theta). We know that:

tan(theta) = sin(theta) / cos(theta)

Step 3: Expressing sin(theta) in terms of cos(theta)

To express sin(theta) in terms of cos(theta), we can use the Pythagorean identity:

sin^2(theta) + cos^2(theta) = 1

Rearranging the equation, we get:

sin^2(theta) = 1 - cos^2(theta)

Taking the square root of both sides, we have:

sin(theta) = √(1 - cos^2(theta))

Step 4: Substituting the expressions for sin(theta) and cos(theta) into the equation for tan(theta)

Substituting the expressions for sin(theta) and cos(theta) into the equation for tan(theta), we get:

tan(theta) = √(1 - cos^2(theta)) / cos(theta)

Step 5: Substituting the expression for cos(theta) from Step 1 into the equation for tan(theta)

Now, we can substitute the expression for cos(theta) from Step 1 into the equation for tan(theta):

tan(theta) = √(1 - (1 / (x^(1/4x)))^2) / (1 / (x^(1/4x)))

Simplifying the expression, we have:

tan(theta) = √(1 - 1 / x^(1/2)) / (1 / x^(1/4x))

Step 6: Simplifying the expression for tan(theta)

To simplify the expression further, we can rationalize the denominator:

tan(theta) = √(x^(1/2) - 1) * (x^(1/4x)) / 1

Simplifying the expression, we get:

tan(theta) = x^(1/4x) * √(x^(1/2) - 1)

Step 7: Expressing sec(theta) * tan(theta) in terms of x

Finally, we can substitute the expression for sec(theta) from the given information:

sec(theta) * tan(theta) = x * (x^(1/4x) * √(x^(1/2) - 1))

Simplifying the expression further, we have:

sec(theta) * tan(theta) = x^(5/4x) * √(x^(1/2) - 1)

Therefore, we have shown that sec(theta) * tan(theta) is equal to x^(5/4x) * √(x^(1/2) -

Sec theta = x+1/4x Sec²theta = (x+1/4x)²1+tan²theta = x²+1/16x²+1/2 tan²theta = x²+1/16x²+1/2-1tan²theta = x²+1/16x²-1/2tan theta = +-(x-1/4x)when tan theta = +(x-1/4x)then, sec theta+tan theta = x+1/4x+x-1/4x => 2xwhen tha theta = -(x-1/4x)then, sec theta +tan theta = x+1/4x-(x-1/4x) =x+1/4x-x+1/4x=> 1/2x So, sec theta +tan theta = 2x or 1/2x