Question

The value of \[{\sin ^2}17.5 + {\sin ^2}72.5\] is equal to
A. \[{\cos ^2}{90^ \circ }\]
B. \[{\tan ^2}{45^ \circ }\]
C. \[{\cos ^2}{30^ \circ }\]
D. \[{\sin ^2}{45^ \circ }\]

Answer 1

Hint: In this question, we find the value of the provided expression. To do that, we use the formula \[\sin \left( {90 - \theta } \right) = \cos \theta \] to turn the second term into a cosine function, simplify it once more, then use the formula \[{\sin ^2}\theta + {\cos ^2}\theta = 1\], and finally consult the trigonometric table to obtain the required outcome.

Formula used:
We have been using the following formulas:
1. \[\sin \left( {90 - \theta } \right) = \cos \theta \]
2. \[{\sin ^2}\theta + {\cos ^2}\theta = 1\]

Complete step-by-step solution:
We are given that \[{\sin ^2}17.5 + {\sin ^2}72.5\]
We are asked to find the value of the given expression.
We know that \[\sin \left( {90 - \theta } \right) = \cos \theta \]
Now we apply the above formula in our expression, and we get
\[{\sin ^2}{17.5^ \circ } + {\sin ^2}\left( {{{90}^ \circ } - {{17.5}^ \circ }} \right)\]
Now by simplifying the above expression, we get
\[{\sin ^2}{17.5^ \circ } + {\cos ^2}{17.5^ \circ }\]
We know that \[{\sin ^2}\theta + {\cos ^2}\theta = 1\]
Now by applying the above formula in our expression, we get
\[{\sin ^2}{17.5^ \circ } + {\cos ^2}{17.5^ \circ } = 1\]
Here, \[\theta = {17.5^ \circ }\]
Now,
 \[
  {\sin ^2}17.5 + {\sin ^2}72.5 = 1 \\
   = {1^2}
 \]
We know that \[\tan {45^ \circ } = 1\]
So, \[{\tan ^2}{45^ \circ } = 1\]
Therefore, the value of \[{\sin ^2}17.5 + {\sin ^2}72.5\]is equal to \[{\tan ^2}{45^ \circ }\]
Hence, option (B) is correct option

Additional information: Trigonometric Identities are equality statements that hold true for all values of the variables in the equation and that use trigonometry functions. There are numerous distinctive trigonometric identities that relate a triangle's side length and angle. Only the right-angle triangle is the exception to the trigonometric identities.

Note: Students should be careful while applying the trigonometric identity in the above problem since they run the risk of making an error and also be attentive when simplifying this identity. So that this minor error does not have an impact on our results.