Composition of Trigonometric Functions and Inverse Trigonometric Functions

Q. Find the value of $\tan \frac{\pi }{8}.$

Q.

If cos(a+b)=4/5, sin(a-b)=5/13 and 0<a, b<pi/4, then tan2a=?

Q.

If ${\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}$ then $x=$

1. $\sqrt{2}$

2. $3$

3. $\sqrt{3}$

4. $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

Q. Prove that:
Sin A Sin(A+2B) - Sin B Sin(B+2A) = Sin(A-B)Sin (A+B)

Q.

Value of sin[1/4arc sin((√63)/8)

Q. The derivative of ${\tan }^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right),$ with respect to $\frac{x}{2},$ where $(x\in (0,\frac{\pi }{2}))$ is:

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

Q.

If sinA+sinB=a and cosA+ cosB= b then the value of cos(A+B) is

Q.

$cot(45°+\theta )cot(45°-\theta )=$

1. $-1$

2. $0$

3. $1$

4. $\infty$

Q. Prove that:

Sin 10° + Sin 20° + Sin 40°+ Sin 50° = Sin 70°+ Sin 80°

Q. Prove that
sin10sin30sin50sin70 = 1/16

Q.

If cos a + cos b + cos c = 0 = sin a + sin b + sin c

cos²a + cos²b + cos²c = ?

Q. Show that ${\sin }^{-1}\frac{12}{13}+{\cos }^{-1}\frac{4}{5}+{\tan }^{-1}\frac{63}{16}=\pi$

Q. Let $f\left(x\right)=2{\tan }^{-1}x$ and $g\left(x\right)=x+2.$ Then the number of integer(s) satisfying $\left(\right(f\circ g\left)\right(x){)}^2-5(f\circ g\left)\right(x)+4>0$ where $x\in (-10,10)$ is

Q. If ${\sin }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{12}{13}={\sin }^{-1}C,$ then $C=$

1. $\frac{8}{65}$
2. $\frac{24}{65}$
3. $\frac{16}{65}$
4. $\frac{56}{65}$

Q.

Prove that

[1/(sec²x-cos²x) + 1/(cosec²x-sin²x)] sin²x cos²x =

(1-sin²x cos²x)/(2+sin²x cos²x)

Q. The value of $cos\left(\frac{1}{2}co{s}^{-1}\left(\frac{1}{8}\right)\right)$ is

1. $-\frac{31}{32}$
2. $\frac{3}{4}$
3. $-\frac{\sqrt{7}}{4}$
4. $-\frac{3}{7}$

Q.

If $A=\frac{(2^{x}cotx)}{\sqrt{x}}$ then $\frac{dA}{dx}=$

1. $\left(\frac{2^x}{\sqrt{x}}\right)[\log 2cotx–\cos e{c}^{2}x–\frac{(cotx)}{2x}]$

2. $\left(\frac{2^x}{x}\right)[\log 2cotx+\cos e{c}^{2}x–\frac{(cotx)}{2x}]$

3. $\left(\frac{2^x}{\sqrt{x}}\right)[\log 2cotx–\cos e{c}^{2}x–(cotx)]$

4. None of these

Q.

Integrate dx/sinx+√3cosx

Q.

cos^{​​​​6} π/16 + cos^{​​​​6} 3π/16 + cos^{​​​​6} 5π/16 + cos^{​​​​6} 7π/16 = ?

Q.

if a = cos 2B + cos 2A. , b= cos 2B - cos 2 A

c= sin 2A+ sin 2B, d= sin = sin2A-sin2B

Then which of the first is true.

(A) a/b= cot(A+B) cot(A-B)

(B) c/d= tan(A+B)/ tan(A-B)

(C ) b/ c= tan(A-B)

(D) none of these

Q. If tanA +secA = 3/2 then find sinA, tanA and secA

Q.

If in $∆\mathrm{ABC},a\mathit{=}\mathit{5}\mathit{,}\mathit{}b\mathit{=}\mathit{4}\mathit{,}\mathit{}A\mathit{=}\frac{\mathit{\pi }}{\mathit{2}}\mathit{+}B$, then $C$

1. is ${\tan }^{-1}\left(\frac{1}{2}\right)$

2. is ${\tan }^{-1}\left(\frac{9}{40}\right)$

3. cannot be evaluated.

4. is ${\tan }^{-1}\left(\frac{1}{9}\right)$

Q. Show that ${\sin }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{8}{17}={\cos }^{-1}\frac{84}{85}$

Q.

$tan\left[se{c}^{-1}\sqrt{1+x^2}\right]=$

1. $\frac{1}{x}$

2. x

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

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

Q. If $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2\text{cosec}x\right)$, then the value of $x$ is:

1. $\frac{3\pi }{4}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{6}$

Q.

Evaluate sin(π\n)+sin(3π\n)+sin(5π\n)+...to n terms

Q. Which of the following pairs of function/functions has same graph?

1. $y=\tan \left({\cos }^{-1}x\right);y=\frac{\sqrt{1-x^2}}{x}$
2. $y=\tan \left({\cot }^{-1}x\right);y=\frac{1}{x}$
   
3. $y=\sin \left({\tan }^{-1}x\right);y=\frac{x}{\sqrt{1+x^2}}$
4. $y=\cos \left({\tan }^{-1}x\right);y=\sin \left({\cot }^{-1}x\right)$

Q. The number of value(s) of $\theta \in [0,2\pi ]$ satisfying the equation $\left({\log }_{\sqrt{5}}\tan \theta \right)\sqrt{{\log }_{\tan \theta }5\sqrt{5}+{\log }_{\sqrt{5}}5\sqrt{5}}=-\sqrt{6}$ is

1. $0$
2. $4$
3. $2$
4. $5$

Q.

How is sinA+sinB+sinC=4 cosA/2 cosB/2 cosC/2

Q. If $y=\left(\frac{{\tan }^{2}2x-{\tan }^{2}x}{1-{\tan }^{2}2x{\tan }^{2}x}\right)\cot 3x,$ then $y^{\prime }\left(\frac{\pi }{4}\right)=$