LFM Pure

jee-main 2021 Q64 Trigonometric Equation Solving via Identities View

If $0 < x , y < \pi$ and $\cos x + \cos y - \cos ( x + y ) = \frac { 3 } { 2 }$, then $\sin x + \cos y$ is equal to:
(1) $\frac { 1 } { 2 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 1 - \sqrt { 3 } } { 2 }$
(4) $\frac { 1 + \sqrt { 3 } } { 2 }$

jee-main 2021 Q66 Simplification of Trigonometric Expressions with Specific Angles View

The value of $\cot \frac { \pi } { 24 }$ is:
(1) $\sqrt { 2 } + \sqrt { 3 } + 2 - \sqrt { 6 }$
(2) $\sqrt { 2 } + \sqrt { 3 } + 2 + \sqrt { 6 }$
(3) $\sqrt { 2 } - \sqrt { 3 } - 2 + \sqrt { 6 }$
(4) $3 \sqrt { 2 } - \sqrt { 3 } - \sqrt { 6 }$

jee-main 2022 Q62 Simplification of Trigonometric Expressions with Specific Angles View

$16 \sin \left( 20 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right) \sin \left( 80 ^ { \circ } \right)$ is equal to
(1) $\sqrt { 3 }$
(2) $2 \sqrt { 3 }$
(3) 3
(4) $4 \sqrt { 3 }$

jee-main 2022 Q64 Simplification of Trigonometric Expressions with Specific Angles View

The value of $2\sin\frac{\pi}{22}\sin\frac{3\pi}{22}\sin\frac{5\pi}{22}\sin\frac{7\pi}{22}\sin\frac{9\pi}{22}$ is equal to:
(1) $\frac{1}{16}$
(2) $\frac{5}{16}$
(3) $\frac{7}{16}$
(4) $\frac{3}{16}$

jee-main 2022 Q66 Simplification of Trigonometric Expressions with Specific Angles View

The value of $2\sin 12^\circ - \sin 72^\circ$ is
(1) $\frac{\sqrt{5}(1 - \sqrt{3})}{4}$
(2) $\frac{1 - \sqrt{5}}{8}$
(3) $\frac{\sqrt{3}(1 - \sqrt{5})}{2}$
(4) $\frac{\sqrt{3}(1 - \sqrt{5})}{4}$

jee-main 2022 Q69 Multi-Step Composite Problem Using Identities View

$\tan \left( 2 \tan ^ { - 1 } \frac { 1 } { 5 } + \sec ^ { - 1 } \frac { \sqrt { 5 } } { 2 } + 2 \tan ^ { - 1 } \frac { 1 } { 8 } \right)$ is equal to:
(1) 1
(2) 2
(3) $\frac { 1 } { 4 }$
(4) $\frac { 5 } { 4 }$

jee-main 2022 Q71 Multi-Step Composite Problem Using Identities View

If $0 < x < \frac { 1 } { \sqrt { 2 } }$ and $\frac { \sin ^ { - 1 } x } { \alpha } = \frac { \cos ^ { - 1 } x } { \beta }$, then a value of $\sin \frac { 2 \pi \alpha } { \alpha + \beta }$ is
(1) $4 \sqrt { 1 - x ^ { 2 } } \left( 1 - 2 x ^ { 2 } \right)$
(2) $4 x \sqrt { 1 - x ^ { 2 } } \left( 1 - 2 x ^ { 2 } \right)$
(3) $2 x \sqrt { 1 - x ^ { 2 } } \left( 1 - 4 x ^ { 2 } \right)$
(4) $4 \sqrt { 1 - x ^ { 2 } } \left( 1 - 4 x ^ { 2 } \right)$

jee-main 2023 Q68 Simplification of Trigonometric Expressions with Specific Angles View

The value of $36 \left( 4 \cos ^ { 2 } 9 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 27 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 81 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 243 ^ { \circ } - 1 \right)$ is
(1) 54
(2) 18
(3) 27
(4) 36

jee-main 2023 Q69 Simplification of Trigonometric Expressions with Specific Angles View

The value of $\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}$ is $\_\_\_\_$.

jee-main 2024 Q64 Addition/Subtraction Formula Evaluation View

For $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$ let $3\sin(\alpha + \beta) = 2\sin(\alpha - \beta)$ and a real number $k$ be such that $\tan\alpha = k\tan\beta$. Then the value of $k$ is equal to
(1) $-5$
(2) $5$
(3) $\frac{2}{3}$
(4) $-\frac{2}{3}$

jee-main 2025 Q7 Telescoping Sum of Trigonometric Terms View

If $\sum _ { r = 1 } ^ { 13 } \left\{ \frac { 1 } { \sin \left( \frac { \pi } { 4 } + ( r - 1 ) \frac { \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { r \pi } { 6 } \right) } \right\} = a \sqrt { 3 } + b , a , b \in \mathbf { Z }$, then $a ^ { 2 } + b ^ { 2 }$ is equal to :
(1) 10
(2) 4
(3) 2
(4) 8

jee-main 2025 Q10 Addition/Subtraction Formula Evaluation View

$\cos \left( \sin ^ { - 1 } \frac { 3 } { 5 } + \sin ^ { - 1 } \frac { 5 } { 13 } + \sin ^ { - 1 } \frac { 33 } { 65 } \right)$ is equal to:
(1) 1
(2) 0
(3) $\frac { 32 } { 65 }$
(4) $\frac { 33 } { 65 }$

jee-main 2025 Q11 Function Analysis via Identity Transformation View

Let the range of the function $f ( x ) = 6 + 16 \cos x \cdot \cos \left( \frac { \pi } { 3 } - x \right) \cdot \cos \left( \frac { \pi } { 3 } + x \right) \cdot \sin 3 x \cdot \cos 6 x , x \in \mathbf { R }$ be $[ \alpha , \beta ]$. Then the distance of the point $( \alpha , \beta )$ from the line $3 x + 4 y + 12 = 0$ is :
(1) 11
(2) 8
(3) 10
(4) 9

taiwan-gsat 2022 Q15 4 marks Function Analysis via Identity Transformation View

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $\frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$. Let this expression be $f(x)$, with domain $\{x \mid 1 < x < 8\}$. Find $f(x)$ and its derivative. (Non-multiple choice question, 4 points)

turkey-yks 2010 Q20 Simplification of Trigonometric Expressions with Specific Angles View

$$\frac{\tan 60^{\circ}}{\sin 20^{\circ}} - \frac{1}{\cos 20^{\circ}}$$
Which of the following is this expression equal to?
A) 4
B) 2
C) 1
D) $\frac{\sqrt{3}}{2}$
E) $\frac{1}{2}$

turkey-yks 2010 Q21 Simplification of Trigonometric Expressions with Specific Angles View

$$\frac{1+\cos 40^{\circ}}{\cos 55^{\circ} \cdot \cos 35^{\circ}}$$
Which of the following is this expression equal to?
A) $\cos 20^{\circ}$
B) $2\cos 20^{\circ}$
C) $4\cos 20^{\circ}$
D) $\cos 40^{\circ}$
E) $2\cos 40^{\circ}$

turkey-yks 2011 Q28 Direct Double Angle Evaluation View

$\cos \mathrm { x } = \frac { - 4 } { 5 }$ Given this, what is $\cos 2 \mathrm { x }$?
A) $\frac { 3 } { 5 }$
B) $\frac { 5 } { 13 }$
C) $\frac { 12 } { 13 }$
D) $\frac { 24 } { 25 }$
E) $\frac { 7 } { 25 }$

turkey-yks 2013 Q25 Trigonometric Equation Solving via Identities View

Given that $\alpha , \beta \in \left[ 0 , \frac { \pi } { 2 } \right]$,
$$\sin ( \alpha - \beta ) = \sin \alpha \cdot \cos \beta$$
Which of the following is true?
A) $\alpha = 0$ or $\beta = \frac { \pi } { 2 }$
B) $\alpha = 0$ or $\beta = \frac { \pi } { 4 }$
C) $\alpha = \frac { \pi } { 2 }$ or $\beta = 0$
D) $\alpha = \frac { \pi } { 2 }$ or $\beta = \frac { \pi } { 2 }$
E) $\alpha = \frac { \pi } { 4 }$ or $\beta = 0$

turkey-yks 2014 Q23 Simplification of Trigonometric Expressions with Specific Angles View

$$\frac { \sin 48 ^ { \circ } } { \sin 16 ^ { \circ } } - \frac { \cos 48 ^ { \circ } } { \cos 16 ^ { \circ } }$$
Which of the following is this expression equal to?
A) $\frac { 3 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 4 } { 3 }$
D) 2
E) 3

turkey-yks 2017 Q51 Trigonometric Identity Proof or Derivation View

For every real number $x$, the number $A$ is defined as $$\sum _ { k = 2 } ^ { 4 } \cos ( 2 k x ) = A$$ Accordingly, $$\sum _ { k = 2 } ^ { 4 } \cos ^ { 2 } ( k x )$$ What is the equivalent of the expression in terms of A?\ A) $A + 2$\ B) $A + 4$\ C) $\frac { \mathrm { A } + 1 } { 2 }$\ D) $\frac { A + 2 } { 2 }$\ E) $\frac { A + 3 } { 2 }$

turkey-yks 2024 Q28 Trigonometric Identity Proof or Derivation View

$$\frac{\cos(2x+y) + \sin(2x-y)}{\cos(2x) + \sin(2x)}$$
Which of the following is the simplified form of this expression?
A) $\cos y - \sin y$ B) $\cos y + \sin y$ C) $\cos x - \sin y$ D) $\sin x - \cos y$ E) $\sin x - \cos x$

turkey-yks 2025 Q29 Geometric Configuration with Trigonometric Identities View

For a triangle $ABC$ with side lengths $|BC| = a$ units, $|AC| = b$ units and $|AB| = c$ units,
$$2a^{2} = 2b^{2} + 2c^{2} + 3bc$$
is satisfied. Let $m(\widehat{BAC}) = x$. What is the value of $\tan x$?
A) $-\frac{\sqrt{2}}{3}$ B) $-\frac{\sqrt{3}}{3}$ C) $-\frac{\sqrt{5}}{3}$ D) $-\frac{\sqrt{6}}{3}$ E) $-\frac{\sqrt{7}}{3}$

turkey-yks 2025 Q31 Geometric Configuration with Trigonometric Identities View

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the acute angles that the lines $d_{1}$ and $d_{2}$ shown make with the $x$-axis are $A$ and $B$ respectively, as shown in the figure.
Accordingly, which of the following is the expression for the ratio $\frac{a}{b}$ in terms of $A$ and $B$?
A) $\frac{\tan A}{\tan B}$ B) $\cot A \cdot \cot B$ C) $\cot A - \tan B$ D) $1 + \cot A \cdot \tan B$ E) $1 - \tan A \cdot \cot B$

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144