Composition of Trigonometric Functions and Inverse Trigonometric Functions

Q.

$\cos ({\tan }^{-1}x)=?$

1. $\sqrt{1+x^2}$

2. $\frac{1}{\sqrt{1+x^2}}$

3. $1+x^2$

4. None of these

Q.

If $\cos (\theta -\alpha )$, $\cos \theta$ and $\cos (\theta +\alpha )$ are in$HP$, then $\cos \theta sec\frac{\alpha }{2}$ equal to

1. $\pm \sqrt{2}$

2. $\pm \sqrt{3}$

3. $\pm \frac{1}{\sqrt{2}}$

4. None of these

Q. Sum to n terms of the series \operatorname{sinθ\operatorname{sin2θ+\operatorname{sin2θ\operatorname{sin3θ+\operatorname{sin3θ\operatorname{sin4θ+...... is equal to

Q.

sin(60° + A) cos(30° - B) + cos(60° + A) sin(30° - B) is equal to

1. cos(A – B)

2. sin(A + B)

3. cos(A + B)

4. sin(A – B)

Q. 28. In Δ ABC if cotß = cotA + cotB + cotC then sin(A-ß) . sin(B-ß) . sin(C-ß) = ………

Q. The smallest value of θ satisfying the equation $\sqrt{3}\left(\cot \mathrm{\theta }+\tan \mathrm{\theta }\right)=4$ is
(a) $2\pi /3$
(b) $\pi /3$
(c) $\pi /6$
(d) $\pi /12$

Q. If cos(α + β )=0, then sin(α -β ) can be equal to (1) cos 2β (2) cos β (3) sin α (4) sin 2α

Q. If sec θ + tan θ = k, cos θ =

(a) $\frac{k^2+1}{2k}$
(b) $\frac{2k}{k^2+1}$
(c) $\frac{k}{k^2+1}$
(d) $\frac{k}{k^2-1}$

Q. 74. If sinα = A Sin(α +β ) . A is not equal to 0. Then tanβ = (1) sinα (1+Acosβ ) / Acosα cosβ (2)sinα (1-Acosβ ) / Acosα cosβ (3)cosα (1-Asinβ ) / Acosα cosβ (4)cosα (1+Asinβ ) / Acosα cosβ

Q. The maximum value of sin² (120° + θ) + sin² (120° − θ) is

(a) 1/2
(b) 3/2
(c) 1/4
(d) 3/4

Q. Prove the following identities (1-17)

${\sec }^4\mathrm{\theta }-{\sec }^2\mathrm{\theta }={\tan }^4\mathrm{\theta }+{\tan }^2\mathrm{\theta }$

Q. $\cos \left({\tan }^{-1}x\right)=$

1. $\sqrt{1+x^2}$
2. $\frac{1}{\sqrt{1+x^2}}$
3. $1+x^2$
4. None of these

Q. If cos (A − B)$=\frac{3}{5}$and tan A tan B = 2, then

(a) $\cos A\cos B=\frac{1}{5}$
(b) $\cos A\cos B=-\frac{1}{5}$
(c) $\sin A\sin B=-\frac{1}{5}$
(d) $\sin A\sin B=-\frac{1}{5}$

Q. The value of $tan[si{n}^{-1}\left(\frac{3}{5}\right)+co{s}^{-1}\left(\frac{3}{\sqrt{13}}\right)]$ is

1. $\frac{6}{17}$
2. $\frac{6}{\sqrt{13}}$
3. $\frac{\sqrt{13}}{5}$
4. $\frac{17}{6}$

Q. If $\mathrm{cosec\theta }-\mathrm{cot\theta }=\frac{1}{2},0<\mathrm{\theta }<\frac{\mathrm{\pi }}{2},$, then cos θ is equal to
(a) $\frac{5}{3}$
(b) $\frac{3}{5}$
(c) $-\frac{3}{5}$
(d) $-\frac{5}{3}$

Q.

prove that sin 50 degrees +sin 10 degres = cos20 degrees

Q. In a triangle ABC , a sin (B - C )+ b sin( C - A)+ c Sin( A- B)=

Q.

If A + B + C = $\pi$, prove that

$\text{cot}\frac{A}{2}+\text{cot}\frac{B}{2}+\text{cot}\frac{C}{2}=\text{cot}\frac{A}{2}\text{cot}\frac{B}{2}\text{cot}\frac{C}{2}$

Q. If 3 sin θ + 5 cos θ = 5, then write the value of 5 sin θ − 3 cos θ.

Q. If $\sin \mathrm{\theta }+\cos \mathrm{\theta }=m$, then prove that

${\sin }^6\mathrm{\theta }+{\cos }^6\mathrm{\theta }=\frac{4-3{\left(m^2-1\right)}^2}{4}$, where $m^2\le 2$

Q. Differentiate (1+cosx) / (1-cosx) with respect to x.

Q. If A + B + C = 180°, then
sec A (cos B cos C − sin B sin C) is equal to

(a) 0
(b) −1
(c) 1
(d) None of these

Q. Prove that : $ta{n}^{2}A-ta{n}^{2}B=\frac{si{n}^{2}A-si{n}^{2}B}{co{s}^{2}A.co{s}^{2}B}$

Q. In ∆ABC, prove the following:
${\sin }^{3}A\cos \left(B-C\right)+{\sin }^{3}B\cos \left(C-A\right)+{\sin }^{3}C\cos \left(A-B\right)=3\sin A\sin B\sin C$

Q. If sin 2 θ + sin 2 ϕ = $\frac{1}{2}$ and cos 2 θ + cos 2 ϕ = $\frac{3}{2}$, then cos² (θ − ϕ) =
(a) $\frac{3}{8}$

(b) $\frac{5}{8}$

(c) $\frac{3}{4}$

(d) $\frac{5}{4}$

Q. The value of $\frac{\cos 3\mathrm{\theta }}{2\cos 2\mathrm{\theta }-1}$ is equal to
(a) $\cos \mathrm{\theta }$
(b) $\sin \mathrm{\theta }$
(c) $\tan \mathrm{\theta }$
(d) none of these

Q. ( sin 75 degrees - sin 15 degrees) /(cos 75 degrees + cos 15 degrees) = 1 / square root of 3.

Q. $\underset{x\to 0}{\lim }\frac{e^{\tan x}-1}{\tan x}$

Q.

The value of $cot(\frac{\pi }{4}-2co{t}^{-1}3)$ is equal to

1. -7

2. 7

3. 6

4. None

Q.

Find the value of tan 75/tan 15