To solve the given expression, we need to simplify it step-by-step. Let's start with the inner expression and move outward.
First, let's evaluate each trigonometric function individually:
1.
tan−1‌The inverse tangent function gives us an angle
θ such that:
tan‌θ=‌ This occurs when
θ is:
θ=‌So,
tan−1‌=‌ 2.
cos−1‌The expression simplifies to:
This occurs when
θ is:
θ=‌So,
cos−1‌=‌ 3.
sin‌−1‌The expression simplifies to:
sin‌−1‌=sin‌−1‌This occurs when
θ is:
θ=‌So,
sin‌−1‌=‌Now we need to add these evaluated inverse trigonometric values:
‌tan−1‌+cos−1‌+sin‌−1‌ Substituting the values, we get:
‌×‌+‌+‌Simplify the expression:
‌+‌+‌ The common denominator here is 12 , so we rewrite each fraction:
‌+‌+‌Add these fractions together:
‌=‌=‌Next, we need to find:
cot(‌)Since
cot‌θ=‌, and
tan(‌) is undefined, so:
cot(‌)=0Finally, we need to find:
sin‌−1[0]This occurs when the angle
θ is:
θ=0So, the value of the given expression is Option D:
0