List of top Mathematics Questions on Trigonometric Identities asked in JEE Main

If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \beta\sqrt{5}}{2}, \] where $ \alpha, \beta \in \mathbb{N} $, then the value of $ \alpha + \beta $ is ___________.

If $\cot x=\dfrac{5}{12}$ for some $x\in(\pi,\tfrac{3\pi}{2})$, then \[ \sin 7x\left(\cos \frac{13x}{2}+\sin \frac{13x}{2}\right) +\cos 7x\left(\cos \frac{13x}{2}-\sin \frac{13x}{2}\right) \] is equal to

The value of $\dfrac{\sqrt{3}\cosec 20^\circ - \sec 20^\circ}{\cos 20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ}$ is equal to

The minimum value of $ \cos^2\theta + 6\sin\theta\cos\theta + 3\sin^2\theta $ is:

If $ \cot\theta = -\dfrac{1}{2\sqrt{2}} $, where $ \theta \in \left( \dfrac{3\pi}{2}, 2\pi \right) $, then the value of \[ \sin\left(\dfrac{15\theta}{2}\right)(\sin 8\theta + \cos 8\theta) + \cos\left(\dfrac{15\theta}{2}\right)(\cos 8\theta - \sin 8\theta) \] is:

The value of $ (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) $ is:

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

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

The value of $ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2 \left( \frac{1}{2} \right)} + 1}{\tan \left( \frac{1}{2} \right)} \right) $ is equal to:

If $ \frac{\pi}{2} \leq x \leq \frac{3\pi}{4} $, then $ \cos^{-1} \left( \frac{12}{13} \cos x + \frac{5}{13} \sin x \right) $ is equal to:

If $ f(x) = \begin{cases} x^3 \sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases} $, then:

If \[\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx = \frac{1}{12} \tan^{-1}(3 \tan x) + \text{constant},\]then the maximum value of $a \sin x + b \cos x$ is:

If $cos 2x-a \sin x=2a-7$ then range of $a$ is:

For $ n \in \mathbb{N} $, if \[ \cot^{-1} 3 + \cot^{-1} 4 + \cot^{-1} 5 + \cot^{-1} n = \frac{\pi}{4}, \] then $ n $ is equal to ______.

If $f(x)$ = $\begin {bmatrix} Cos x& -sinx & 0\\sinx & cos x& 0\\0&0&1 \end {bmatrix} $Statement I $⇒ f(x).f(y) = f(x+y)$ Statement II $⇒f(-x) =0 $ is invertible

$sin^{-1}\bigg(sin\frac{2π}{3}\bigg)+cos^{-1}\bigg(cos\frac{7π}{6}\bigg)+tan^{-1}\bigg(tan\frac{3π}{4}\bigg)$ is equal to:

If $cos^{-1}(\frac{y}{2})=log_e(\frac{x}{5})^5, |y| < 2$ then:

The angle of elevation of a jet plane from a point A on the ground is $60^{\circ}$. After a flight of 20 seconds at the speed of $432 km /$ hour, the angle of elevation changes to $30^{\circ} .$ If the jet plane is flying at a constant height, then its height is :

Given $\frac{b+c}{11} = \frac{c+a}{12}= \frac{a+b}{13} $ for a $ \Delta ABC$ with usual notation. If $ \frac{\cos A}{\alpha} = \frac{\cos B}{\beta} = \frac{\cos C}{\gamma} $ , then the ordered triad $( \alpha, \beta, \gamma)$ has a value :

If the angle of elevation of a cloud from a point $P$ which is $25\, m$ above a lake be $30^\circ$ and the angle of depression of reflection of the cloud in the lake from $P$ be $60^\circ$, then the height of the cloud (in meters) from the surface of the lake is :

In a $\Delta ABC$. with usual notation, match the items in List - I with the items in List - II and choose the correct option.

List I		List II
(A)	$r_1r_2\sqrt{\bigg(\frac{4R-r_1-r_2}{r_1+r_2}\bigg)}$	1	$b$
(B)	$\frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}$	2	$a^2,b^2,c^2 are \;in \;AP$
(C)	$\frac{a}{c}=\frac{sin(A-B)}{sin(B-C)}$	3	$\triangle$
(D)	$bc\;cos^2\frac{A}{2}$	4	$R\; r_1r_2r_3$
		5	$s(s-a)$

A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min. for the angle of depression of the car to change from $30^{\circ}$ to $45^{\circ}$ ; then after this, the time taken (in min.) by the car to reach the foot of the tower, is :

Let $S$ be the set of all solutions of the equation $\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^2}\right)=\pi$, $x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$ Then $\displaystyle\sum_{x \in S} 2 \sin ^{-1}\left(x^2-1\right)$ is equal to

In a ΔPQR, if 3sinP + 4cosQ = 6 and 4sinQ + 3cosP = 1, then the angle R is equal to:

For x ∈ (0, π), the equation sin x + 2sin2x −sin3x = 3 has

List I		List II
(A)	\(r_1r_2\sqrt{\bigg(\frac{4R-r_1-r_2}{r_1+r_2}\bigg)}\)	1	\(b\)
(B)	\(\frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}\)	2	\(a^2,b^2,c^2 are \;in \;AP\)
(C)	\(\frac{a}{c}=\frac{sin(A-B)}{sin(B-C)}\)	3	\(\triangle\)
(D)	\(bc\;cos^2\frac{A}{2}\)	4	\(R\; r_1r_2r_3\)
		5	\(s(s-a)\)