The number of elements in the set x [0, 180] : (x + 100) = (x + 50) x (x

Q68 jee_main_2025_08_april_evening Simplification of Trigonometric Expressions

The value of

⁻¹ ( 1 + ^ 2 (2) - 1 (2)) - ⁻¹ ( 1 + ^ 2 ((1)/(2)) + 1 ((1)/(2)))

is equal to

A. $π -(5)/(4)$ $\pi -\frac{5}{4}$
B. $π -(3)/(2)$ $\pi -\frac{3}{2}$
C. $π +(3)/(2)$ $\pi +\frac{3}{2}$
D. $π +(5)/(2)$ $\pi +\frac{5}{2}$

Solution

Related Formula

\sqrt(1+ ²θ) = | θ|

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:

| 2| = - 2

(- 2 - 1)/( 2) = (-1 - 2)/( 2) = - 1

Step 2: Simplify Second Exponent Operand

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

| (1/2)| = (1/2)

( (1/2) + 1)/( (1/2)) = (1 + (1/2))/( (1/2)) = (1/4)

Step 3: Combine Structural Terms

Apply inverse mapping functions carefully:

⁻¹(- 1) - ⁻¹( (1)/(4)) = (π - 1) - (1)/(4) = π - (5)/(4)

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

Q52 jee_main_2025_29_jan_evening Trigonometric Identities

If

x + ² x = 1

,

x in (0, (π)/(2))

, then

( ¹² x + ¹² x) + 3 ( ¹⁰ x + ¹⁰ x + ⁸ x + ⁸ x) + ( ⁶ x + ⁶ x)

is equal to

A. $4$ $4$
B. $3$ $3$
C. $2$ $2$
D. $1$ $1$

Solution

Related Formula

Fundamental identities:

² x + ² x = 1

x = ( x)/( x)

Algebraic identity for a perfect cube:

(A + B)³ = A³ + 3A²B + 3AB² + B³

Core Logic

Given equation:

x + ² x = 1 x = 1 - ² x = ² x

Dividing both sides by $² x$ $\cos^2 x$:

( x)/( ² x) = 1 x x = 1 x = 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:

( ¹² x + ¹² x) + 3( ¹⁰ x + ¹⁰ x + ⁸ x + ⁸ x) + ( ⁶ x + ⁶ x)

= 2 ¹² x + 6 ¹⁰ x + 6 ⁸ x + 2 ⁶ x

= 2[ ¹² x + 3 ¹⁰ x + 3 ⁸ x + ⁶ x]

Step 2: Apply the Cubic Identity

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

= 2[( ⁴ x + ² x)³]

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:

= 2[( ² x + x)³]

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

Q jee_main_2025_28_jan_morning Sum of Inverse Trigonometric Functions

( ⁻¹(3)/(5) + ⁻¹(5)/(13) + ⁻¹(33)/(65))

is equal to:

A. 1
B. 0
C. $(33)/(65)$ $\frac{33}{65}$
D. $(32)/(65)$ $\frac{32}{65}$

Solution

Related Formula

Standard tangent identity sum format:

⁻¹ x + ⁻¹ y = ⁻¹ ((x+y)/(1-xy))

Core Logic

Convert all components into tangent mappings: $⁻¹(3)/(5) = ⁻¹(3)/(4)$ $⁻¹(5)/(13) = ⁻¹(5)/(12)$ $⁻¹(33)/(65) = ⁻¹(33)/(56)$

Step 1: Evaluating the Mapped Component Sum

Summing the first two components:

⁻¹(3)/(4) + ⁻¹(5)/(12) = ⁻¹(((3)/(4) + (5)/(12))/(1 - (15)/(48))) = ⁻¹(56)/(33)

Step 2: Applying Cofunction Complements

Notice that $⁻¹(33)/(56) = ⁻¹(56)/(33)$ . Combining everything inside the function:

( ⁻¹(56)/(33) + ⁻¹(56)/(33)) = ((π)/(2)) = 0

Pattern Recognition

Look for reciprocal fractional identities across matching inverse blocks—they easily merge using the $⁻¹ x + ⁻¹ x = (π)/(2)$ identity.

Chapter Mix

Class 12 Maths: Inverse Trigonometric Functions

Q60 jee_main_2025_03_april_morning Trigonometric Equations

The number of solutions of the equation

2x + 3 x = π

,

xin[-2π, 2π] - \pm(π)/(2), \pm(3π)/(2)

is:

A. 6
B. 5
C. 4
D. 3

Solution

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.

Trigonometric Equations diagram for Q60 - JEE Main 2025 Morning

Core Logic

Rearrange terms to group equations into known standard graphing profiles:

3 x = π - 2x x = (π)/(3) - (2x)/(3)

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 $[-2π, 2π]$ $[-2\pi, 2\pi]$.

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:

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

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

Q jee_main_2025_04_april_evening Summation of Series

The sum of the infinite series

⁻¹ ((7)/(4)) + ⁻¹ ((1 9)/(4)) + ⁻¹ ((3 9)/(4)) + ⁻¹ ((6 7)/(4)) + .

is :-

A. $(π)/(2) + ⁻¹((1)/(2))$
B. $(π)/(2) - ⁻¹((1)/(2))$
C. $(π)/(2) + ⁻¹((1)/(2))$
D. $(π)/(2) - ⁻¹((1)/(2))$

Solution

Related Formula

The general transformation formula for a difference of tangents is:

⁻¹x - ⁻¹y = ⁻¹((x-y)/(1+xy))

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:

Numerator = 4n² + 3

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

Tₙ = ⁻¹((4n² + 3)/(4)) = ⁻¹((4)/(4n² + 3))

Step 1: Rewriting the general term for telescoping sum

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

Tₙ = ⁻¹((1)/(n² + (3)/(4))) = ⁻¹((1)/(1 + (n² - (1)/(4))))

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

Tₙ = ⁻¹(((n + (1)/(2)) - (n - (1)/(2)))/(1 + (n + (1)/(2))(n - (1)/(2))))

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

Tₙ = ⁻¹(n + (1)/(2)) - ⁻¹(n - (1)/(2))

Step 2: Telescoping summation

Expanding the sum up to $n$ $n$ terms:

Sₙ = Σk=1ⁿ Tk = [ ⁻¹((3)/(2)) - ⁻¹((1)/(2))] + [ ⁻¹((5)/(2)) - ⁻¹((3)/(2))] + + [ ⁻¹(n + (1)/(2)) - ⁻¹(n - (1)/(2))]

All intermediate terms cancel out, leaving:

Sₙ = ⁻¹(n + (1)/(2)) - ⁻¹((1)/(2))

Step 3: Infinite limit evaluation

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

S_\infty = n \to \infty [ ⁻¹(n + (1)/(2)) - ⁻¹((1)/(2))] = (π)/(2) - ⁻¹((1)/(2))

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

The number of elements in the set x in [0, 180°] : (x + 100°) = (x + 50°) x (x - 50°)$\{x \in [0, 180^{\circ}] : \tan(x + 100^{\circ}) = \tan(x + 50^{\circ}) \tan x \tan(x - 50^{\circ})\}$ is

Solution & Explanation

Related Formula

Core Logic

Step 1: Componendo and Dividendo

Step 2: Trigonometric Simplification

Step 3: Solving for x

Step 4: Finding Specific Roots

Pattern Recognition

Chapter Mix

Reference Study Guides

More Trigonometric Equations Previous-Year Questions — Page 5

Solution

Related Formula

Core Logic

Step 1: Simplify First Exponent Operand

Step 2: Simplify Second Exponent Operand

Step 3: Combine Structural Terms

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Simplify the Expression

Step 2: Apply the Cubic Identity

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Evaluating the Mapped Component Sum

Step 2: Applying Cofunction Complements

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Point analysis across branches

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Rewriting the general term for telescoping sum

Step 2: Telescoping summation

Step 3: Infinite limit evaluation

Pattern Recognition

Chapter Mix

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The number of elements in the set $x in [0, 180°] : (x + 100°) = (x + 50°) x (x - 50°)$ is