Advertisements
Advertisements
Question
Prove the following result:
`sin^-1 12/13+cos^-1 4/5+tan^-1 63/16=pi`
Solution
LHS = `sin^-1 12/13+cos^-1 4/5+tan^-1 63/16`
`=tan^-1 (12/13)/sqrt(1-144/169)+tan^-1 sqrt(1-16/25)/(4/5)+tan^-1 63/16` `[becausesin^-1x=tan^-1 x/sqrt(1-x^2) and cos^-1x=tan^-1 sqrt(1-x^2)/x]`
`=tan^-1 (12/13)/(5/13)+tan^-1 (3/5)/(4/5)+tan^-1 63/16`
`=tan^-1 12/5+tabn^-1 3/4+tan^-1 63/16`
`=pi+tan^-1((12/5+3/4)/(1-12/5xx3/4))+tan^-1 63/16` `[because tan^-1x+tan^-1y=pi+tan^-1((x+y)/(1-xy))]`
`=pi+tan^-1((63/20)/(-16/20))+tan^-1 63/16`
`=pi+tan^-1 (-63)/16+tan^-1 63/16`
`=pi-tan^-1 63/16+tan^-1 63/16`
= π = RHS
APPEARS IN
RELATED QUESTIONS
Write the value of `tan(2tan^(-1)(1/5))`
Solve the equation for x:sin−1x+sin−1(1−x)=cos−1x
Find the domain of `f(x) =2cos^-1 2x+sin^-1x.`
`sin^-1(sin (5pi)/6)`
Evaluate the following:
`cos^-1{cos (13pi)/6}`
Evaluate the following:
`tan^-1(tan (9pi)/4)`
Evaluate the following:
`cot^-1{cot ((21pi)/4)}`
Write the following in the simplest form:
`tan^-1sqrt((a-x)/(a+x)),-a<x<a`
Evaluate the following:
`cot(cos^-1 3/5)`
Evaluate: `sin{cos^-1(-3/5)+cot^-1(-5/12)}`
`sin(sin^-1 1/5+cos^-1x)=1`
`sin^-1x=pi/6+cos^-1x`
Find the value of `tan^-1 (x/y)-tan^-1((x-y)/(x+y))`
Solve the following equation for x:
`tan^-1(2+x)+tan^-1(2-x)=tan^-1 2/3, where x< -sqrt3 or, x>sqrt3`
`sin^-1 63/65=sin^-1 5/13+cos^-1 3/5`
`(9pi)/8-9/4sin^-1 1/3=9/4sin^-1 (2sqrt2)/3`
Solve the following:
`sin^-1x+sin^-1 2x=pi/3`
Solve the equation `cos^-1 a/x-cos^-1 b/x=cos^-1 1/b-cos^-1 1/a`
`2tan^-1 3/4-tan^-1 17/31=pi/4`
Show that `2tan^-1x+sin^-1 (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.
Find the value of the following:
`cos(sec^-1x+\text(cosec)^-1x),` | x | ≥ 1
Solve the following equation for x:
`2tan^-1(sinx)=tan^-1(2sinx),x!=pi/2`
Write the difference between maximum and minimum values of sin−1 x for x ∈ [− 1, 1].
If −1 < x < 0, then write the value of `sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))`
Write the value of sin−1
\[\left( \sin( -{600}°) \right)\].
Evaluate sin
\[\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right)\]
Write the value of cos \[\left( 2 \sin^{- 1} \frac{1}{2} \right)\]
Show that \[\sin^{- 1} (2x\sqrt{1 - x^2}) = 2 \sin^{- 1} x\]
Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]
Write the principal value of \[\sin^{- 1} \left\{ \cos\left( \sin^{- 1} \frac{1}{2} \right) \right\}\]
If \[\cos^{- 1} \frac{x}{a} + \cos^{- 1} \frac{y}{b} = \alpha, then\frac{x^2}{a^2} - \frac{2xy}{ab}\cos \alpha + \frac{y^2}{b^2} = \]
sin \[\left\{ 2 \cos^{- 1} \left( \frac{- 3}{5} \right) \right\}\] is equal to
If \[\cos^{- 1} x > \sin^{- 1} x\], then
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find \[\frac{dy}{dx}\] When \[\theta = \frac{\pi}{3}\] .
Find the domain of `sec^(-1) x-tan^(-1)x`
The equation sin-1 x – cos-1 x = cos-1 `(sqrt3/2)` has ____________.
