Let f 1 x -sin -1cossin 2 x - f 2 x -cos -1sincos 2 x - f 3 x -sin -1coscos 2 x and f 4 x -cos -1sinsin 2 x - Thenwhich of the following is are CORRECT-A- -i-14 f i x -2 -2B- -i-14-1 i f i x -0C- -i-14-1- i - 3- f i x -0- where - denotes the greatest integer function D- f 1 x - f 2 x -cos 2 x -0

The correct options are A ∑_i=1^4f_i(x) = 2π 2 B ∑_i=1^4( 1)^i f_i(x) = 0 C ∑_i=1^4 ( 1)^[i/3] f_i(x) = 0, where [.] denotes the greatest integer function n ...

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Byju's Answer

Standard XII

Mathematics

Composition of Inverse Trigonometric Functions and Trigonometric Functions

Let f 1 x =si...

Question

Let $f_{1} (x) = {sin}^{- 1} (cos ({sin}^{2} x)),$ $f_{2} (x) = {cos}^{- 1} (sin ({cos}^{2} x)),$ $f_{3} (x) = {sin}^{- 1} (cos ({cos}^{2} x))$ and $f_{4} (x) = {cos}^{- 1} (sin ({sin}^{2} x)) .$ Then which of the following is (are) CORRECT?

4 \sum i = 1 f_{i} (x) = 2 π - 2

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4 \sum i = 1 (- 1)^{i} f_{i} (x) = 0

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4 \sum i = 1 (- 1)^{[i / 3]} f_{i} (x) = 0,

where

[.]

denotes the greatest integer function

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f_{1} (x) > f_{2} (x) \Rightarrow cos 2 x > 0

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Solution

The correct options are
A $4 \sum i = 1 f_{i} (x) = 2 π - 2$
B $4 \sum i = 1 (- 1)^{i} f_{i} (x) = 0$
C $4 \sum i = 1 (- 1)^{[i / 3]} f_{i} (x) = 0,$ where $[.]$ denotes the greatest integer function
D $f_{1} (x) > f_{2} (x) \Rightarrow cos 2 x > 0$
$f_{1} (x) = \frac{π}{2} - {sin}^{2} x$
$f_{2} (x) = \frac{π}{2} - {cos}^{2} x$
$f_{3} (x) = \frac{π}{2} - {cos}^{2} x$
$f_{4} (x) = \frac{π}{2} - {sin}^{2} x$

$f_{1} (x) = f_{4} (x)$ and $f_{3} (x) = f_{2} (x)$
$4 \sum i = 1 f_{i} (x) = f_{1} (x) + f_{2} (x) + f_{3} (x) + f_{4} (x)$
$= 2 (\frac{π}{2} - {sin}^{2} x + \frac{π}{2} - {cos}^{2} x)$
$= 2 (π - 1)$

$4 \sum i = 1 (- 1)^{i} f_{i} (x) = - f_{1} (x) + f_{2} (x) - f_{3} (x) + f_{4} (x) = 0$

$4 \sum i = 1 (- 1)^{[i / 3]} f_{i} (x) = f_{1} (x) + f_{2} (x) - f_{3} (x) - f_{4} (x) = 0$

$f_{1} (x) > f_{2} (x)$
$\Rightarrow - {sin}^{2} x > - {cos}^{2} x$
$\Rightarrow {cos}^{2} x - {sin}^{2} x > 0$
$\Rightarrow cos 2 x > 0$

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Let f1(x)=sin−1(cos(sin2x)), f2(x)=cos−1(sin(cos2x)), f3(x)=sin−1(cos(cos2x)) and f4(x)=cos−1(sin(sin2x)). Then which of the following is (are) CORRECT?

Let $f_{1} (x) = {sin}^{- 1} (cos ({sin}^{2} x)),$ $f_{2} (x) = {cos}^{- 1} (sin ({cos}^{2} x)),$ $f_{3} (x) = {sin}^{- 1} (cos ({cos}^{2} x))$ and $f_{4} (x) = {cos}^{- 1} (sin ({sin}^{2} x)) .$ Then which of the following is (are) CORRECT?