Question

If we have an expression as \[y={{\sec }^{-1}}\left[ \csc x \right]+{{\csc }^{-1}}\left[ \sec x \right]+{{\sin }^{-1}}\left[ \cos x \right]+{{\cos }^{-1}}\left[ \sin x \right]\] then \[\dfrac{dy}{dx}\] is
1. \[0\]
2. \[2\]
3. \[-2\]
4. \[-4\]

Answer 1

Hint: Here we need to find the differentiation of the given expression. As we know that to first find the differentiation of the expression we will need to first simplify the expression to be able to find the difference. Because for this expression we can not apply any directly known formula. Now here we know that cosec is the supplement of sec hence we can use that property to simplify the first part and the second part of this expression. Similarly we also know that sin and cos are also the supplementary angles to each other and as we did for cosec and sec we can do the same for sine and cos. Once simplified we just need to differentiate.

Complete step-by-step solution:
Now to start we must start with the given expression which is
\[y={{\sec }^{-1}}\left[ \csc x \right]+{{\csc }^{-1}}\left[ \sec x \right]+{{\sin }^{-1}}\left[ \cos x \right]+{{\cos }^{-1}}\left[ \sin x \right]\]
Now to simplify this so you can find the differentiate of this expression first you must know that cosec is the supplementary angle to secant which we can say means that (or can be expressed by);
\[\csc x=\sec (90{}^\circ -x)\]
Now similarly even secant is the supplementary angle to cosec which can be expressed by
\[\sec x=\csc (90{}^\circ -x)\]
Also for sine and cosine we know both of these angles are supplementary to each other meaning that we can express both of them are
\[\sin x=\cos (90{}^\circ -x)\] ; \[\cos x=\sin (90{}^\circ -x)\]
Now substituting these expressions in the main equations we get that
\[y={{\sec }^{-1}}\left[ \sec (90-x) \right]+{{\csc }^{-1}}\left[ \csc (90-x) \right]+{{\sin }^{-1}}\left[ \sin (90-x) \right]+{{\cos }^{-1}}\left[ \cos (90-x) \right]\]
Now since these are inverse trigonometric functions and we know their properties we can write this as
\[y=90-x+90-x+90-x+90-x\]
Simplifying
\[y=4(90-x)\]
Opening the bracket we get
\[y=360-4x\]
Now differentiating it we get the answer we need which is
\[\dfrac{dy}{dx}=-4\]
Hence the differentiate of \[y={{\sec }^{-1}}\left[ \csc x \right]+{{\csc }^{-1}}\left[ \sec x \right]+{{\sin }^{-1}}\left[ \cos x \right]+{{\cos }^{-1}}\left[ \sin x \right]\] gives us the answer which is \[\dfrac{dy}{dx}=-4\]

Note: Two angles are supplementary to each other if when we add both of the angles we get a straight angle that is \[\theta =180{}^\circ \] . Something one should be cautious about in questions like this is that the angle inside the inverse and function is in the range of the inverse functions input or else we would get a different answer