Conditional Identities

Q.

If 15 $si{n}^4\alpha$ + 10 $co{s}^4\alpha$ = 6 then the value of 8 $cose{c}^6\alpha$ - 27 $se{c}^6\alpha$ is __.

Q.

Prove that,

(i) $cos\alpha +cos\beta +cos\gamma +cos(\alpha +\beta +\gamma )=4cos\left(\frac{\alpha +\beta }{2}\right).cos\left(\frac{\beta +\gamma }{2}\right).cos\left(\frac{\gamma +\alpha }{2}\right).$

(ii) If $tanx=\frac{b}{a},$ then $\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}=\frac{2cosx}{\sqrt{cos2x}}.$

Q.

If A + B + C = $\pi$ then $ta{n}^2\frac{A}{2}$ + $ta{n}^2\frac{B}{2}$ + $ta{n}^2\frac{C}{2}$ is always

1. 1

2. 1

3. = 0

4. = 1

Q.

The minimum value of cos(cos x) is

1. -1

2. 0

3. cos1

4. -cos1

Q.

In a $△ABC,cosA=\frac{3}{5}andcosB=\frac{5}{13}$ The value of cot C can be

1. 

2. 

3. 

4. None of these

Q. 34. The expression " cos3x + sin3x +(2sin2x-3)(sinx- cosx)", is positive for all x in the range?

Q.

If $a{\cos }^3\alpha +3a\cos \alpha {\sin }^2\alpha =m$ and

$a{\sin }^3\alpha +3a{\cos }^2\alpha \sin \alpha =n$, Then $(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}$

is equal to

1. $2a^2$

2. $2a^{\frac{1}{3}}$

3. $2a^{\frac{2}{3}}$

4. $2a^3$

Q.

Find the value of tan 3A in terms of tanA & tan2A

1. 

2. 

3. 

4. 

Q.

If $A+B+C={180}^{\circ }$. Find ${\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C$ =

1. $-1-4\cos A\cos B\cos C$

2. $1+2\cos A\cos B\cos C$

3. $2+2\cos A\cos B\cos C$

4. $4\sin A\sin B\sin C$

Q. If in a $∆ABC,\tan A+\tan B+\tan C=0$, then $\cot A\cot B\cot C=$
(a) 6
(b) 1
(c) $\frac{1}{6}$
(d) none of these

Q.

Solve $tanx+tan2x+tan3x=tanxtan2xtan3x.$

OR

prove that $cot\frac{\pi }{24}=\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}.$

Q. In $△ABC$,
$\tan A.\tan B.\tan C=\tan A+\tan B+\tan C$

1. False
2. True

Q.

If A = $\{\frac{1}{x}:xϵN\text{and}x<8\}\text{and}B=\{\frac{1}{2x}:xϵ\text{and}x\le 4\}$, find B- A

Q.

If $A+B+C={180}^{\circ }$. Find ${\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C$ =

1. $-1-4\cos A\cos B\cos C$

2. $1+2\cos A\cos B\cos C$

3. $2+2\cos A\cos B\cos C$

4. $4\sin A\sin B\sin C$

Q. What is the value of cot 4A?
Also, what is the value of cot 7 1/2 (ie, cot7.5)? need a quick reply

Q. Let $A+B+C=\pi$ and $\alpha ={\sin }^3(B+C)\cdot \sin (2C+A),\beta ={\sin }^3(A+C)\cdot \sin (2A+B),\gamma ={\sin }^3(A+B)\cdot \sin (2B+C)$
are roots of the cubic equation $x^3+a{x}^2+bx+c=0$, then the value of $a$ is

1. $0$
2. $\sin A\sin B\sin C$
3. $1$
4. $\cos A\cos B\cos C$

Q. In a $△ABC$, if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A\sin B\sin C=\frac{3+\sqrt{3}}{8}$, then

The value of $\tan A\tan B+\tan B\tan C+\tan C\tan A$ is

1. $5-4\sqrt{3}$
2. $5+4\sqrt{3}$
3. $6+\sqrt{3}$
4. $6-\sqrt{3}$

Q. If $\alpha =cos\left(\frac{8\pi }{11}\right)+isin\left(\frac{8\pi }{11}\right),thenRe(\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5)$ is equal to

1. 0
2. None of these
3. 
4. 

Q.

If $a{\cos }^3\alpha +3a\cos \alpha {\sin }^2\alpha =m$ and

$a{\sin }^3\alpha +3a{\cos }^2\alpha \sin \alpha =n$, Then $(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}$

is equal to

1. $2a^2$

2. $2a^{\frac{1}{3}}$

3. $2a^{\frac{2}{3}}$

4. $2a^3$

Q. If $a{\cos }^3\alpha +3a\cos \alpha {\sin }^2\alpha =m$ and $a{\sin }^3\alpha +3a{\cos }^2\alpha \sin \alpha =n$, then $(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}=$

1. $2a^2$
2. $2a^{\frac{1}{3}}$
3. $2a^{\frac{2}{3}}$
4. $2a^3$

Q. If in triangle ABC, tan A + tan B + tan C = 6 and tan A tan B = 2, then the triangle is

1. Right angled isosceles
2. Aacute angled
3. Aacute angled isosceles
4. equilateral

Q. If $\frac{3\mathrm{\pi }}{4}<\mathrm{\alpha }<\mathrm{\pi },\mathrm{then}\sqrt{2\cot \mathrm{\alpha }+\frac{1}{{\sin }^2\mathrm{\alpha }}}$ is equal to
(a) 1 − cot α
(b) 1 + cot α
(c) −1 + cot α
(d) −1 −cot α

Q. In a $△ABC$, if $\cos A+\cos B+\cos C=\frac{3}{2}$, then the triangle is

1. an isosceles
2. an equilateral
3. a scalene
4. a right angled

Q.

find the value of

cos 10⁰ cos 50⁰ cos 30⁰ cos 70⁰

Q.

Find the value of 1 + $\alpha$ + ${\alpha }^2$ + ${\alpha }^3$ +.............. ${\alpha }^{n-1}$. If ${\alpha }^k$ = $cos\frac{2\pi k}{n}+isin\frac{2\pi k}{n}$,

K = 0, 1, 2, ..........n-1

__

Q. Value of integration from π/6 to π/3 sinx /x lies between A) 1 & π/2 Β) π/6 & π/3 C) 1/3 & π/6 D) 1/3 & π/3

Q. If $\cos x=\frac{1}{2}\left(a+\frac{1}{a}\right),$ and $\cos 3x=\lambda \left(a^3+\frac{1}{a^3}\right)$, then $\lambda =$
(a) $\frac{1}{4}$

(b) $\frac{1}{2}$

(c) 1

(d) none of these

Q.

If $aco{s}^3\alpha +3acos\alpha si{n}^2\alpha$ = m and

$asi{n}^3\alpha +3aco{s}^2\alpha sin\alpha$ = n, Then $(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}$

is equal to

1. 

2. 

3. 

4. 

Q. If ${\cos }^6\alpha +{\sin }^6\alpha +k{\sin }^{2}2\alpha =1\forall \alpha \in \left(0,\pi /2\right)$, then k is

1. 
2. 
3. 
4. 

Q. Let $A,B,C$ are three angles such that
$\sin A+\sin B+\sin C=0,$ then the value of $\frac{\sin A.\sin B.\sin C}{\sin 3A+\sin 3B+\sin 3C}$ (wherever defined) is

1. $12$
2. $-12$
3. $-\frac{1}{12}$
4. $\frac{1}{12}$