Question

How do you verify the identity \[\sin \left( {90 + A} \right) = \cos A\]

Answer 1

Hint: To solve the question given above, use the trigonometric identities. Remember that while verifying trigonometric questions always make use of the identities and formulas. For this question we will use the \[A + B\] identity where \[\sin \left( {A + B} \right) = \sin A\cos B + cosA\sin B\].

Formula used: In order to verify the above question, we will take help of the following formula:
\[\sin \left( {A + B} \right) = \sin A\cos B + cosA\sin B\].

Complete step-by-step answer:
In the above question we are given that: \[\sin \left( {90 + A} \right) = \cos A\].
To verify that \[\sin \left( {90 + A} \right) = \cos A\], we will use the formula that has been mentioned above to expand the terms.
\[\sin \left( {A + B} \right) = \sin A\cos B + cosA\sin B\]
On expanding, we get:
\[\sin 90\cos A + \cos 90\sin A = \cos A\]
Now, we are well aware of the fact that \[\sin 90^\circ = 1\] and \[\cos 90^\circ = 0\].
On equating these values in the above equation, we get:
\[
  1\left( {\cos A} \right) + 0\left( {\sin A} \right) = \cos A \\
   \Rightarrow \cos A = \cos A \\
 \]
So, we get that \[L.H.S = R.H.S\].
Hence proved that \[\sin \left( {90 + A} \right) = \cos A\].

Note: The trigonometric functions are real functions that link a right-angled triangle's angle to two-side length ratios. They are commonly used in all geodetic sciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many others. They're one of the most basic periodic functions. The sine, cosine, and tangent are the most commonly used trigonometric functions in modern mathematics. The cosecant, secant, and cotangent are their reciprocals, which are less commonly used. Each of these six trigonometric functions has an inverse function (known as an inverse trigonometric function) and a hyperbolic function counterpart.