Question

In a triangle \[ABC\], find the value of the expression\[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right)\;\].
A. \[1\]
B. \[\dfrac{c}{a}\]
C. \[\dfrac{a}{c}\]
D. None of these

Answer 1

Hint: In the given question, we need to find the value of an expression\[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right)\;\]. For this, we need to use the property of a triangle such that the sum of all the angles of a triangle is \[{180^ \circ }\]. Also, we will use the following trigonometric identities to get the desired result.

Formula used:
 The following trigonometric properties are used for solving this question.
\[\sin \left( {{{180}^ \circ } - \theta } \right) = \sin \theta \] and \[\cos ec\theta = \dfrac{1}{{\sin \theta }}\]

Complete step by step solution:
We know that in \[\Delta ABC\], \[\angle A + \angle B + \angle C = {180^ \circ }\]
Now, we will find the value of an expression \[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right)\;\].
Here, we know that \[\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right)\; = \sin \left( {B + C} \right)\]
Thus, we get
\[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right) = cosec{\rm{ }}A{\rm{ }}\left( {sin\left( {B + C} \right)} \right)\;\]
But \[\angle B + \angle C = {180^ \circ } - \angle A\]
\[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right) = cosec{\rm{ }}A{\rm{ }}\left( {sin\left( {{{180}^ \circ } - A} \right)} \right)\]
Also, we know that \[\sin \left( {{{180}^ \circ } - A} \right) = \sin A\]
So, we get
\[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right) = cosec{\rm{ }}A{\rm{ }}\left( {\sin A} \right)\]
But \[\cos ecA = \dfrac{1}{{\sin A}}\]
Finally, we get
\[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right) = \dfrac{1}{{\sin A}}{\rm{ }}\left( {\sin A} \right)\]
This gives \[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right) = 1\]
Hence, In a triangle \[ABC\], the value of the expression \[cosec{\rm{ }}A{\rm{ }}\left( {sinB{\rm{ }}cosC{\rm{ }} + {\rm{ }}cosB{\rm{ }}sinC} \right)\;\]is \[1\].
Therefore, the correct option is (A).

Additional information: Trigonometric identities are equalities in trigonometry that are valid for any value of the occurring variables at which both halves of the equation are specified. In simple words, the trigonometric identities are equalities using trigonometric functions that remain true for any value of the variables involved, hence defining both halves of the equality.
Sin, cos, and tan are the three primary trigonometric ratios whereas sec, cosec, and cot are the secondary trigonometric ratios. Also, all the trigonometric identities are associated with the right angled triangle. The trigonometric identities are useful to simplify complex and tricky trigonometric expressions.

Note: Many students make mistakes in solving calculation parts and applying trigonometric identities. This is the only way through which we can solve the example in the simplest way. To use proper trigonometric identities is necessary for solving trigonometric problems as this makes them simple.