If 10 ⁴θ + 15 ⁴θ = 6$10\sin^4\theta + 15\cos^4\theta = 6$, then the value of (27 ⁶θ + 8 ⁶θ)/(16 ⁸θ)$\frac{27\csc^6\theta + 8\sec^6\theta}{16\sec^8\theta}$ is:

Related Formula

Trigonometric Sine Identity:

Core Logic

Let ⁻¹x = θ x = θ$\sin^{-1}x = \theta \implies x = \sin\theta$ and √(1-x²) = θ$\sqrt{1-x^2} = \cos\theta$. Given constraint -(1)/(2) < x < 1√(2) -(π)/(6) < θ < (π)/(4)$-\frac{1}{2} < x < \frac{1}{\sqrt{2}} \implies -\frac{\pi}{6} < \theta < \frac{\pi}{4}$.

Substitute parameter representations into expression:

Step 1: Check Principal Bounds

Evaluate bounds for arguments: since -(π)/(6) < θ < (π)/(4)$-\frac{\pi}{6} < \theta < \frac{\pi}{4}$:

This lies completely within the principal value branch of ⁻¹x$\sin^{-1}x$, which is [-(π)/(2), (π)/(2)]$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Therefore, ⁻¹( (θ + (π)/(6))) = θ + (π)/(6)$\sin^{-1}\left(\sin\left(\theta + \frac{\pi}{6}\right)\right) = \theta + \frac{\pi}{6}$.

Step 2: Final Form

Substituting back θ = ⁻¹x$\theta = \sin^{-1}x$:

Pattern Recognition

Always check primary interval bounds when stripping inverse operators. If the arguments exceed bounds, quadrant mapping transformations must be performed.

Chapter Mix

Class 12 Mathematics: Inverse Trigonometric Functions

Related Formula

Product-to-sum formula and triple angle identity are:

Core Logic

Given equation:

Multiplying by 2:

Using product-to-sum on the first term:

Step 1: Structural Rearrangement

Simplifying through standard trigonometric transformation equations leads directly to:

Hence, either (3θ) = 0$\sin(3\theta) = 0$ or \sin\left(\frac{9\theta}{2}\right) = 0.

Step 2: Finding Roots in the Interval

Interval given:$.

Step 2: Finding Roots in the Interval

Interval given: $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right].

Case A:$.

Case A: $sin(3\theta) = 0 \implies 3\theta = n\pi \implies \theta = \frac{n\pi}{3}Values inside interval:$ Values inside interval: $\left\{-\frac{pi}{3}, 0, \frac{\pi}{3}\right\}(3 solutions).

Case B:$ (3 solutions).

Case B: $sin\left(\frac{9\theta}{2}\right) = 0 \implies \frac{9\theta}{2} = m\pi \implies \theta = \frac{2m\pi}{9}Values inside interval:$ Values inside interval: $\left\{-\frac{4\pi}{9}, -\frac{2\pi}{9}, 0, \frac{2\pi}{9}, \frac{4\pi}{9}\right\}. Since$. Since $0is already counted, this gives 4 unique additional solutions.

Total unique solutions =$ is already counted, this gives 4 unique additional solutions.

Total unique solutions = $3 + 4 = 7.

Pattern Recognition

Transforming powers like$.

Pattern Recognition

Transforming powers like $\cos^3 x$ back into simple multiple-angle terms linearizes trigonometric equations instantly for direct factoring.

Chapter Mix

Class 11 Mathematics: Trigonometry

Related Formula

The standard conversion between ⁻¹(x)$\cot^{-1}(x)$ and ⁻¹(x)$\tan^{-1}(x)$ depends on the sign of x$x$:

Core Logic

Simplify the interior terms algebraic representations:

Step 1: Convert to Inverse Tangent terms

Since α > β > γ > 0$\alpha > \beta > \gamma > 0$:

• (1+αβ)/(α-β) > 0 ⇒ ⁻¹((1+αβ)/(α-β)) = ⁻¹((α-β)/(1+αβ))$\frac{1+\alpha\beta}{\alpha-\beta} > 0 \Rightarrow \cot^{-1}\left(\frac{1+\alpha\beta}{\alpha-\beta}\right) = \tan^{-1}\left(\frac{\alpha-\beta}{1+\alpha\beta}\right)$
• (1+βγ)/(β-γ) > 0 ⇒ ⁻¹((1+βγ)/(β-γ)) = ⁻¹((β-γ)/(1+βγ))$\frac{1+\beta\gamma}{\beta-\gamma} > 0 \Rightarrow \cot^{-1}\left(\frac{1+\beta\gamma}{\beta-\gamma}\right) = \tan^{-1}\left(\frac{\beta-\gamma}{1+\beta\gamma}\right)$
• (1+αγ)/(γ-α) < 0$\frac{1+\alpha\gamma}{\gamma-\alpha} < 0$ (since γ - α < 0$\gamma - \alpha < 0$) ⇒ ⁻¹((1+αγ)/(γ-α)) = π + ⁻¹((γ-α)/(1+αγ))$\Rightarrow \cot^{-1}\left(\frac{1+\alpha\gamma}{\gamma-\alpha}\right) = \pi + \tan^{-1}\left(\frac{\gamma-\alpha}{1+\alpha\gamma}\right)$

Step 2: Telescopic Sum Evaluation

Apply the difference identity for arctan, ⁻¹((x-y)/(1+xy)) = ⁻¹x - ⁻¹y$\tan^{-1}\left(\frac{x-y}{1+xy}\right) = \tan^{-1}x - \tan^{-1}y$ :

All variables cancel symmetrically leaving:

= π$= \pi$

Pattern Recognition

The sign trap is the most vital component of this question. The ordering α > β > γ > 0$\alpha > \beta > \gamma > 0$ means the last term contains a denominator with a negative difference (γ - α)$(\gamma - \alpha)$, introducing the +π$+\pi$ offset according to the principal range of ⁻¹(x)$\cot^{-1}(x)$.

Chapter Mix

Class 12 Mathematics: Inverse Trigonometric Functions

Related Formula

Logarithmic addition property: (a) + (b) = (ab)$\log(a) + \log(b) = \log(ab)$. Double angle sine formula: 2 x x = 2x$2\sin x \cos x = \sin 2x$.

Step 1: Simplify Set A

Combine the logarithmic elements:

Since π² ≈ 9.87$\pi^2 \approx 9.87$, (8)/(π²) ≈ 0.81$\frac{8}{\pi^2} \approx 0.81$, which is less than 1. Plotting | 2x| = (8)/(π²)$|\sin 2x| = \frac{8}{\pi^2}$ over the specified range x in (0, π)$x \in (0, \pi)$ yields exactly 4 real intersection points.

Hence, n(A) = 4$n(A) = 4$.

Step 2: Simplify Set B

Let √(x) = t$\sqrt{x} = t$ where t ≥ 0$t \ge 0$. The equation becomes:

Case I: If t < 2$t < 2$ :

Case II: If t > 2$t > 2$ :

Hence, set B = 0, 1, 9, 16$B = \{0, 1, 9, 16\}$, giving n(B) = 4$n(B) = 4$.

Step 3: Calculate Union

Since all elements of set A are non-integral angles in (0, π)$(0, \pi)$ and elements of set B are pure integers, the sets are completely disjoint (A B =$A \cap B = \emptyset$).

Pattern Recognition

Always separate functions into disjoint numeric domains (e.g., angles vs. whole integers) to conclude unions without performing tedious tracking of individual values manually.

Chapter Mix

Class 11 Physics: Trigonometric Functions Class 11 Mathematics: Sets