If for some $\alpha, \beta$; $\alpha \leq \beta$, $\alpha + \beta = 8$ and $\sec^2(\tan^{-1} \alpha) + \csc^2(\cot^{-1} \beta) = 36$, then $\alpha^2 + \beta$ is

Step 1: Simplify the trigonometric expression

We use the identities $\sec^2 x = 1 + \tan^2 x$ and $\csc^2 x = 1 + \cot^2 x$. Let $x = \tan^{-1} \alpha$ and $y = \cot^{-1} \beta$. Then $\tan x = \alpha$ and $\cot y = \beta$. Substituting these into the given equation simplifies the expression.

Step 2: Substitute and simplify

After applying the identities, we substitute $\tan(\tan^{-1} \alpha) = \alpha$ and $\cot(\cot^{-1} \beta) = \beta$. This leads to a simplified algebraic equation involving $\alpha^2$ and $\beta^2$.

Step 3: Use the sum of roots identity

We are given $\alpha + \beta = 8$ and we found $\alpha^2 + \beta^2 = 34$. We can use the algebraic identity for the square of a sum to find the product $\alpha\beta$.

Step 4: Calculate the product $\alpha\beta$

Substitute the known values into the identity. This allows us to solve for $2\alpha\beta$ and then for $\alpha\beta$.

Step 5: Form a quadratic equation for $\alpha$ and $\beta$

We can form a quadratic equation whose roots are $\alpha$ and $\beta$ using the sum and product of the roots. The general form is $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$.

Step 6: Solve the quadratic equation

Factor the quadratic equation to find the possible values for $x$. These values represent $\alpha$ and $\beta$.

Step 7: Determine $\alpha$ and $\beta$ and calculate $\alpha^2 + \beta$

Since it is given that $\alpha \leq \beta$, we assign $\alpha = 3$ and $\beta = 5$. Finally, we calculate the required expression $\alpha^2 + \beta$.

More from this chapter

• cos ( sin^(-1) (3)/(8) + sin^(-1) (5)/(13) + sin^(-1) (33)/(65)) is equal to:
• Considering the principal values of the inverse trigonometric functions, sin^(-1) ((√3)/(2)x + (1)/(2)√(1-x²)), -(1)/(2) < x < (1)/(√2) is equal to:
• If α > β > > 0, then the expression cot^(-1) β + ((1 + β^5))/((α - β)) + cot^(-1) + ((1 + ²))/((β -)) + cot^(-1) α + ((1 + α²))/(( - α)) is equal to:
• The value of cot^(-1) ( (√(1 + tan²(2)) - 1)/(tan(2))) - cot^(-1) ( (√(1 + tan²((1)/(2))) + 1)/(tan((1)/(2)))) is equal to:
• Solve for x: sin⁻¹(x) + sin⁻¹(√(1-x²)) = π/2 for x ∈ [0,1].
• Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of 16((sec^(-1) x)² + (csc^(-1) x)²) is:
• If y = cos((π)/(3) + cos^(-1)(x)/(2)), then (x - y)² + 3y² is equal to:
• If (π)/(2) ≤ x ≤ (3π)/(4), then cos^(-1) ( (12)/(13) cos x + (5)/(13) sin x) is equal to

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