LFM Pure

jee-advanced 2009 Q32 Trigonometric Equation Constraint Deduction View

If
$$\frac { \sin ^ { 4 } x } { 2 } + \frac { \cos ^ { 4 } x } { 3 } = \frac { 1 } { 5 } ,$$
then
(A) $\quad \tan ^ { 2 } x = \frac { 2 } { 3 }$
(B) $\quad \frac { \sin ^ { 8 } x } { 8 } + \frac { \cos ^ { 8 } x } { 27 } = \frac { 1 } { 125 }$
(C) $\quad \tan ^ { 2 } x = \frac { 1 } { 3 }$
(D) $\frac { \sin ^ { 8 } x } { 8 } + \frac { \cos ^ { 8 } x } { 27 } = \frac { 2 } { 125 }$

jee-main 2011 Q66 Extremal Value of Trigonometric Expression View

If $A=\sin^{2}x+\cos^{4}x$, then for all real $x$
(1) $\frac{13}{16}\leq\mathrm{A}\leq 1$
(2) $1\leq A\leq 2$
(3) $\frac{3}{4}\leq\mathrm{A}\leq\frac{13}{16}$
(4) $\frac{3}{4}\leq\mathrm{A}\leq 1$

jee-main 2013 Q68 Trigonometric Identity Simplification View

The expression $\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}$ can be written as:
(1) $\tan A + \cot A$
(2) $\sec A + \operatorname{cosec} A$
(3) $\sin A \cos A + 1$
(4) $\sec A \operatorname{cosec} A + 1$

jee-main 2018 Q67 Trigonometric Equation Constraint Deduction View

If $\tan A$ and $\tan B$ are the roots of the quadratic equation, $3 x ^ { 2 } - 10 x - 25 = 0$ then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cdot \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is
(1) 25
(2) - 25
(3) - 10
(4) 10

jee-main 2019 Q65 Trigonometric Equation Constraint Deduction View

If $\sin ^ { 4 } \alpha + 4 \cos ^ { 4 } \beta + 2 = 4 \sqrt { 2 } \sin \alpha \cos \beta , \alpha , \beta \in [ 0 , \pi ]$, then $\cos ( \alpha + \beta ) - \cos ( \alpha - \beta )$ is equal to
(1) - 1
(2) $- \sqrt { 2 }$
(3) $\sqrt { 2 }$
(4) 0

jee-main 2019 Q66 Trigonometric Identity Simplification View

The value of $\cos ^ { 2 } 10 ^ { \circ } - \cos 10 ^ { \circ } \cos 50 ^ { \circ } + \cos ^ { 2 } 50 ^ { \circ }$ is
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 4 } + \cos 20 ^ { \circ }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 3 } { 2 } \left( 1 + \cos 20 ^ { \circ } \right)$

jee-main 2019 Q67 Trigonometric Identity Simplification View

The value of $\sin 10 ^ { \circ } \sin 30 ^ { \circ } \sin 50 ^ { \circ } \sin 70 ^ { \circ }$ is:
(1) $\frac { 1 } { 36 }$
(2) $\frac { 1 } { 16 }$
(3) $\frac { 1 } { 18 }$
(4) $\frac { 1 } { 32 }$

jee-main 2021 Q63 Power-Sum Evaluation via Trigonometric Constraint View

If $15 \sin ^ { 4 } \alpha + 10 \cos ^ { 4 } \alpha = 6$, for some $\alpha \in R$, then the value of $27 \sec ^ { 6 } \alpha + 8 \operatorname { cosec } ^ { 6 } \alpha$ is equal to:
(1) 350
(2) 500
(3) 400
(4) 250

jee-main 2022 Q84 Trigonometric Identity Simplification View

If $\sin ^ { 2 } \left( 10 ^ { \circ } \right) \sin \left( 20 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right) \sin \left( 50 ^ { \circ } \right) \sin \left( 70 ^ { \circ } \right) = \alpha - \frac { 1 } { 16 } \sin \left( 10 ^ { \circ } \right)$, then $16 + \alpha ^ { - 1 }$ is equal to $\_\_\_\_$.

jee-main 2023 Q64 Trigonometric Identity Simplification View

$96 \cos\frac{\pi}{33} \cos\frac{2\pi}{33} \cos\frac{4\pi}{33} \cos\frac{8\pi}{33} \cos\frac{16\pi}{33}$ is equal to
(1) 3
(2) 1
(3) 4
(4) 2

jee-main 2023 Q68 Trigonometric Identity Simplification View

Let $f ( \theta ) = 3 \left( \sin ^ { 4 } \left( \frac { 3 \pi } { 2 } - \theta \right) + \sin ^ { 4 } ( 3 \pi + \theta ) \right) - 2 \left( 1 - \sin ^ { 2 } 2 \theta \right)$ and $S = \left\{ \theta \in [ 0 , \pi ] : f ^ { \prime } ( \theta ) = - \frac { \sqrt { 3 } } { 2 } \right\}$. If $4 \beta = \sum _ { \theta \in S } \theta$ then $f ( \beta )$ is equal to
(1) $\frac { 11 } { 8 }$
(2) $\frac { 5 } { 4 }$
(3) $\frac { 9 } { 8 }$
(4) $\frac { 3 } { 2 }$

jee-main 2024 Q64 Trigonometric Equation Constraint Deduction View

If $\sin x = - \frac { 3 } { 5 }$, where $\pi < x < \frac { 3 \pi } { 2 }$, then $80 \left( \tan ^ { 2 } x - \cos x \right)$ is equal to
(1) 108
(2) 109
(3) 18
(4) 19

jee-main 2024 Q77 Trigonometric Inequality Proof View

Between the following two statements: Statement I : Let $\vec { a } = \hat { i } + 2 \hat { j } - 3 \hat { k }$ and $\vec { b } = 2 \hat { i } + \hat { j } - \hat { k }$. Then the vector $\vec { r }$ satisfying $\vec { a } \times \vec { r } = \vec { a } \times \vec { b }$ and $\vec { a } \cdot \vec { r } = 0$ is of magnitude $\sqrt { 10 }$. Statement II : In a triangle $A B C , \cos 2 A + \cos 2 B + \cos 2 C \geq - \frac { 3 } { 2 }$.
(1) Statement I is incorrect but Statement II is correct.
(2) Both Statement I and Statement II are correct.
(3) Statement I is correct but Statement II is incorrect.
(4) Both Statement I and Statement II are incorrect.

jee-main 2025 Q18 Trigonometric Identity Simplification View

The value of $\left( \sin 70 ^ { \circ } \right) \left( \cot 10 ^ { \circ } \cot 70 ^ { \circ } - 1 \right)$ is
(1) $2/3$
(2) 1
(3) 0
(4) $3/2$

jee-main 2025 Q20 Power-Sum Evaluation via Trigonometric Constraint View

If $\sin x + \sin ^ { 2 } x = 1 , x \in \left( 0 , \frac { \pi } { 2 } \right)$, then $\left( \cos ^ { 12 } x + \tan ^ { 12 } x \right) + 3 \left( \cos ^ { 10 } x + \tan ^ { 10 } x + \cos ^ { 8 } x + \tan ^ { 8 } x \right) + \left( \cos ^ { 6 } x + \tan ^ { 6 } x \right)$ is equal to :
(1) 4
(2) 1
(3) 3
(4) 2

taiwan-gsat 2020 Q1 6 marks Ordering or Comparing Trigonometric Expressions View

Given $45^{\circ} < \theta < 50^{\circ}$, and let $a = 1 - \cos^{2}\theta$, $b = \frac{1}{\cos\theta} - \cos\theta$, $c = \frac{\tan\theta}{\tan^{2}\theta + 1}$. Regarding the relative sizes of the three values $a$, $b$, $c$, select the correct option.
(1) $a < b < c$
(2) $a < c < b$
(3) $b < a < c$
(4) $b < c < a$
(5) $c < a < b$

turkey-yks 2010 Q19 Trigonometric Identity Simplification View

$$\frac{(\sin x - \cos x)^{2}}{\cos x} + 2\sin x$$
Which of the following is this expression equal to?
A) $\frac{1}{\cos x}$
B) $\frac{1}{\sin x}$
C) $1$
D) $\arcsin x$
E) $\arccos x$

turkey-yks 2014 Q34 Triangle Trigonometric Relation View

In the regular pentagon ABCDE shown, K and L are the midpoints of line segments AB and DA, respectively.
Given this, what is the measure of angle LKB in degrees?
A) 105
B) 108
C) $\mathbf { 1 2 0 }$
D) 126
E) 135

turkey-yks 2017 Q30 Triangle Trigonometric Relation View

ABC is an isosceles triangle $\mathrm { D } \in [ \mathrm { BC } ] , \mathrm { B } \in [ \mathrm { AE } ]$ $| \mathrm { AB } | = | \mathrm { BC } |$ $| \mathrm { AC } | = | \mathrm { AD } | = | \mathrm { DE } |$ $\mathrm { m } ( \widehat { \mathrm { ADB } } ) = 111 ^ { \circ }$ $m ( \widehat { B D E } ) = x$
Accordingly, how many degrees is $x$?
A) 15 B) 18 C) 21 D) 24 E) 27

turkey-yks 2018 Q27 Trigonometric Identity Simplification View

$$\frac { \cot \left( 34 ^ { \circ } \right) \cdot \sin \left( 44 ^ { \circ } \right) } { \sin \left( 22 ^ { \circ } \right) \cdot \sin \left( 56 ^ { \circ } \right) }$$
What is the equivalent of this expression?
A) $2 \cot \left( 22 ^ { \circ } \right)$ B) $2 \cos \left( 56 ^ { \circ } \right)$ C) $4 \sin \left( 44 ^ { \circ } \right)$ D) $4 \cos \left( 34 ^ { \circ } \right)$ E) $4 \tan \left( 56 ^ { \circ } \right)$

turkey-yks 2018 Q29 Trigonometric Identity Simplification View

Below are shown a semicircle with center O and radius 1 unit, and right triangles OAB and ODC. Points A and C lie on both the triangle OAB and the semicircle.
Accordingly, $$\frac { | \mathrm { AB } | + | \mathrm { BC } | } { | \mathrm { CD } | + | \mathrm { DA } | }$$
What is the equivalent of this ratio in terms of x?
A) $\sin x$ B) $\tan x$ C) $\cot x$ D) $\csc x$ E) $\sec x$

turkey-yks 2019 Q33 Trigonometric Identity Simplification View

Two right triangles $A B C$ and $B C D$ with one side coinciding are drawn as shown in the figure, and the resulting two regions are painted yellow and blue.
$$\mathrm { m } ( \widehat { \mathrm { DCA } } ) = \mathrm { m } ( \widehat { \mathrm { BAC } } ) = \mathrm { x }$$
Accordingly, what is the expression in terms of x for the ratio of the area of the yellow region to the area of the blue region?
A) $\sin 2 x$
B) $\cos 2 x$
C) $\sin ^ { 2 } x$
D) $\cot ^ { 2 } x$
E) $\csc ^ { 2 } x$

turkey-yks 2020 Q27 Trigonometric Identity Simplification View

$$\frac{2\tan x - \sin(2x)}{\sin^2 x}$$
What is the simplified form of this expression?
A) $2\tan x$
B) $\tan(2x)$
C) $2\cos x$
D) $\cos(2x)$
E) 1

turkey-yks 2020 Q31 Triangle Trigonometric Relation View

In the figure, the line segments $[OA]$ and $[OD]$ intersect perpendicularly.
Accordingly, the ratio of the area of triangle OAB to the area of triangle OCD in terms of $\alpha$ is which of the following?
A) $\tan \alpha$
B) $\cot \alpha$
C) $\tan^2 \alpha$
D) $\cot^2 \alpha$
E) $\sec^2 \alpha$

turkey-yks 2023 Q27 Ordering or Comparing Trigonometric Expressions View

Let $a \in \left(\frac{3\pi}{4}, \pi\right)$,
$$\begin{aligned} & x = \sin(2a) \cdot \tan(a) \\ & y = \cos(2a) \cdot \cot(2a) \\ & z = \sin(a) \cdot \cot(2a) \end{aligned}$$
Given these equalities.
Accordingly, what are the signs of $\mathbf{x}$, $y$ and $\mathbf{z}$ respectively?
A) $+, +, -$ B) $+, -, -$ C) $-, -, -$ D) $-, +, +$ E) $-, -, +$

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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