COMPASSLearning

Talk to us Register Login

JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Inverse Trigonometric Functions

cos (sin^{- 1} \frac{3}{5} + sin^{- 1} \frac{5}{13} + sin^{- 1} \frac{33}{65})

is equal to :

(A)1
(B)0✓ Correct
(C) $\frac{33}{65}$
(D) $\frac{32}{65}$

Answer: (B)

Solution

Let

α = sin^{- 1} \frac{3}{5}, β = sin^{- 1} \frac{5}{13}, γ = sin^{- 1} \frac{33}{65}

From these:

tan α = \frac{3}{4}, tan β = \frac{5}{12}

, and since

cos γ = 1 - (\frac{33}{65})^{2} = \frac{56}{65}

,

tan γ = \frac{33}{56}

.

tan (α + β) = \frac{\frac{3}{4} + \frac{5}{12}}{1 - \frac{3}{4} \cdot \frac{5}{12}} = \frac{14/12}{33/48} = \frac{56}{33}

Since

tan (α + β) \cdot tan γ = \frac{56}{33} \cdot \frac{33}{56} = 1

, and both

α + β

and

γ

lie in

(0, \frac{π}{2})

, it follows that

α + β + γ = \frac{π}{2}

cos (α + β + γ) = cos \frac{π}{2} = 0

JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Inverse Trigonometric Functions →

More JEE Main questions on Inverse Trigonometric Functions

If the domain of the function $f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1)$ is the interval $[α, β]$ , then $α + 2 β$ is equal to :
JEE Main 2026View solution →
The number of solutions of $tan^{- 1} 4 x + tan^{- 1} 6 x = \frac{π}{6}$ , where $- \frac{1}{2 6} < x < \frac{1}{2 6}$ is equal to
JEE Main 2026View solution →
If the domain of the function $f (x) = sin^{- 1} (\frac{1}{x ^{2} - 2 x - 2})$ , is $(- \infty, α] \cup [β, γ] \cup [δ, \infty)$ , then $α + β + γ + δ$ is equal to
JEE Main 2026View solution →
If $k = tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} (\frac{2}{3})) + tan (\frac{1}{2} sin^{- 1} (\frac{2}{3}))$ then the number of solutions of the equation …
JEE Main 2026View solution →
Considering the principal values of inverse trigonometric functions, the value of the expression $tan (2 sin^{- 1} (\frac{2}{13}) - 2 cos^{- 1} (\frac{3}{10}))$ is equal to :
JEE Main 2026View solution →
Let [ • ] denote the greatest integer function. If the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 2 [ x ]}{3})$ is $[α, β]$ , then $12 (α + β)$ is equal to :
JEE Main 2026View solution →