The number of solutions of the equation 2 θ (θ)/(2) + (5 θ)/(2) = 2 ^ 3 (5 θ)/(2)$\cos 2 \theta \cos \frac {\theta}{2} + \cos \frac {5 \theta}{2} = 2 \cos^ {3} \frac {5 \theta}{2}$ in [ - (π)/(2), (π)/(2) ]$\left[ - \frac {\pi}{2}, \frac {\pi}{2} \right]$ is:

Related Formula

Core Logic

Track angular positions across quadrants accurately. Evaluate positive/negative absolute value signs based on component radian locations before reducing formulas.

Step 1: Simplify First Exponent Operand

For tracking segment angle height θ = 2$\theta = 2$ radians (Quadrant II), cosine terms switch below zero:

Step 2: Simplify Second Exponent Operand

For tracking segment angle height θ = 1/2$\theta = 1/2$ radian (Quadrant I), expressions remain positive:

Step 3: Combine Structural Terms

Apply inverse mapping functions carefully:

Pattern Recognition

Radian values like 2 sit over 90^°$90^\circ$ but beneath 180^°$180^\circ$. Missing quadrant validation tags is a common trap in inverse identity tracking.

Chapter Mix

Class 12 Mathematics: Inverse Trigonometric Functions

Related Formula

Fundamental identities:

Algebraic identity for a perfect cube:

Core Logic

Given equation:

Dividing both sides by ² x$\cos^2 x$:

Step 1: Simplify the Expression

Since x = x$\tan x = \cos x$, we can substitute x$\tan x$ with x$\cos x$ throughout the given expression:

Step 2: Apply the Cubic Identity

Notice that the expression inside the brackets matches the expansion of a perfect cube:

Since ² x = x$\cos^2 x = \sin x$, it follows that ⁴ x = ² x$\cos^4 x = \sin^2 x$. Substituting these back in:

We know from the problem statement that x + ² x = 1$\sin x + \sin^2 x = 1$. Therefore:

= 2(1)³ = 2$= 2(1)^3 = 2$

Pattern Recognition

When given x + ² x = 1$\sin x + \sin^2 x = 1$, the substitution ² x = x$\cos^2 x = \sin x$ or x = x$\tan x = \cos x$ is a classic identity trick. Recognizing binomial coefficients (1, 3, 3, 1)$(1, 3, 3, 1)$ immediately signals to condense into a full cube structure.

Chapter Mix

Class 11 Mathematics: Trigonometric Functions

Related Formula

Standard tangent identity sum format:

Core Logic

Convert all components into tangent mappings: ⁻¹(3)/(5) = ⁻¹(3)/(4)$\sin^{-1}\frac{3}{5} = \tan^{-1}\frac{3}{4}$ ⁻¹(5)/(13) = ⁻¹(5)/(12)$\sin^{-1}\frac{5}{13} = \tan^{-1}\frac{5}{12}$ ⁻¹(33)/(65) = ⁻¹(33)/(56)$\sin^{-1}\frac{33}{65} = \tan^{-1}\frac{33}{56}$

Step 1: Evaluating the Mapped Component Sum

Summing the first two components:

Step 2: Applying Cofunction Complements

Notice that ⁻¹(33)/(56) = ⁻¹(56)/(33)$\tan^{-1}\frac{33}{56} = \cot^{-1}\frac{56}{33}$. Combining everything inside the function:

Pattern Recognition

Look for reciprocal fractional identities across matching inverse blocks—they easily merge using the ⁻¹ x + ⁻¹ x = (π)/(2)$\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$ identity.

Chapter Mix

Class 12 Maths: Inverse Trigonometric Functions

Related Formula

Intersection method for transcendental configurations: f(x) = g(x)$f(x) = g(x)$ Plot both curves separately to observe distinct intersection markers inside target domain spans.

Core Logic

Rearrange terms to group equations into known standard graphing profiles [cite: 1321]: 3 x = π - 2x x = (π)/(3) - (2x)/(3)$$3\tan x = \pi - 2x \implies \tan x = \frac{\pi}{3} - \frac{2x}{3}$$ [cite: 1321]

Plot the linear equation line y = (π)/(3) - (2x)/(3)$y = \frac{\pi}{3} - \frac{2x}{3}$ along with multiple period tracks of the trigonometric function y = x$y = \tan x$ within the interval [-2π, 2π]$[-2\pi, 2\pi]$[cite: 613, 1321].

Step 1: Point analysis across branches

The line has a negative slope and passes through (0, π/3)$(0, \pi/3)$ and (3π/2, 0)$(3\pi/2, 0)$. Looking across distinct interval chunks separated by asymptotes[cite: 613]:

• Branch 1 (-2π, -(3π)/(2))$\left(-2\pi, -\frac{3\pi}{2}\right)$: 1 intersection
• Branch 2 (-(3π)/(2), -(π)/(2))$\left(-\frac{3\pi}{2}, -\frac{\pi}{2}\right)$: 1 intersection
• Branch 3 (-(π)/(2), (π)/(2))$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$: 1 intersection near origin
• Branch 4 ((π)/(2), (3π)/(2))$\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$: 1 intersection
• Branch 5 ((3π)/(2), 2π)$\left(\frac{3\pi}{2}, 2\pi\right)$: 1 intersection

Counting all distinct points across valid domains gives 5 solutions total[cite: 1337].

Pattern Recognition

A linear curve intersecting tangent asymptote branches will cut exactly once through every continuous range slice unless the line is strictly horizontal or parallel to asymptotes.

Chapter Mix

Class 11 Mathematics: Trigonometric Functions

Related Formula

The general transformation formula for a difference of tangents is:

Core Logic

Let the general term of the series be Tₙ$T_n$. Examining the numerators (7, 19, 39, 67,$7, 19, 39, 67, \dots$):

The differences between consecutive terms are 12, 20, 28,$12, 20, 28, \dots$, which forms an arithmetic progression with a common difference of 8$8$.

Thus, the general term for the sequence in the numerator can be found using difference methods:

Therefore, the n$n$-th term Tₙ$T_n$ is:

Step 1: Rewriting the general term for telescoping sum

Divide the numerator and denominator inside the argument by 4$4$:

Factorize n² - (1)/(4)$n^2 - \frac{1}{4}$ as a difference of squares:

Using the difference formula for ⁻¹$\tan^{-1}$:

Step 2: Telescoping summation

Expanding the sum up to n$n$ terms:

All intermediate terms cancel out, leaving:

Step 3: Infinite limit evaluation

Taking the limit as n → ∞$n \to \infty$:

Pattern Recognition

Whenever you see an infinite series involving ⁻¹$\cot^{-1}$ or ⁻¹$\tan^{-1}$, try to rearrange the denominator into the form 1 + xy$1 + xy$ and check if the numerator matches x - y$x - y$ to set up a standard telescoping structure.

Chapter Mix

Class 12 Mathematics: Inverse Trigonometric Functions Class 11 Mathematics: Sequences and Series