Question

What will the value of \[\cos A + \sin \left( {{{270}^\circ } + A} \right) - \sin \left( {{{270}^\circ } - A} \right) + \cos \left( {{{180}^\circ } + A} \right)\]?
A. \[ - 1\]
B. \[0\]
C. \[1\]
D. None of these

Answer 1

Hint: We know that to solve the given equation first simplify these values \[\sin \left( {{{270}^\circ } + A} \right),\sin \left( {{{270}^\circ } - A} \right),\cos \left( {{{180}^\circ } + A} \right)\] in the form of cosine and then substitute the values in the given equation to get the desired result.

Formula Used: We have used the formula of periodic identities that are given below
 \[
  \sin ((3\pi /2) + x) = - \cos x \\
  \sin ((3\pi /2) - x) = - \cos x \\
  \cos (\pi + x) = - \cos x \\
 \]

Complete step-by-step solution:
We are given an equation that is \[\cos A + \sin \left( {{{270}^\circ } + A} \right) - \sin \left( {{{270}^\circ } - A} \right) + \cos \left( {{{180}^\circ } + A} \right)\].
Now we apply the formula of periodic identities in the given equation, and we get
\[
  \cos A + \sin \left( {{{270}^\circ } + A} \right) - \sin \left( {{{270}^\circ } - A} \right) + \cos \left( {{{180}^\circ } + A} \right) = \cos A - \cos A - \left( { - \cos A} \right) + \left( { - \cos A} \right) \\
   = \cos A - \cos A + \cos A - \cos A \\
   = 0
 \]
Therefore, the value of \[\cos A + \sin \left( {{{270}^\circ } + A} \right) - \sin \left( {{{270}^\circ } - A} \right) + \cos \left( {{{180}^\circ } + A} \right) = 0\]

Hence, the value of the equation \[\cos A + \sin \left( {{{270}^\circ } + A} \right) - \sin \left( {{{270}^\circ } - A} \right) + \cos \left( {{{180}^\circ } + A} \right)\] is \[0\].

 So, option B is correct

Additional information: The sides of a right-angled triangle are used in sin-cos formulas. Sin and cos, along with the tan function, are basic trigonometric functions. The sine of an angle is equal to the opposite side to hypotenuse ratio, whereas the cosine of an angle is equal to the adjacent side to hypotenuse ratio. Trigonometry has six functions: sin, cos, tan, cosec, sec, and tan. Using a unit circle, we can determine the sign of the trigonometric function. All functions are positive in the first quadrant, only sin and cosec are positive in the second quadrant, tan and cot are positive in the third quadrant, and cos and sec are positive in the fourth quadrant.

Note: Many students made miscalculations while writing the values of the sine and cosine formula so make sure about the formula and also solve the question with the help of the formula. Also, they do not know the exact values of sine or cosine in different quadrants and where to write the sign of positive or negative.