The Value of \( \tan x \sin\left(\frac{\pi}{2}+x\right)\cos\left(\frac{\pi}{2}-x\right) \)
Question
Find the value of
\[ \tan x \cdot \sin\left(\frac{\pi}{2}+x\right)\cdot \cos\left(\frac{\pi}{2}-x\right) \]
(a) \(1\)
(b) \(-1\)
(c) \(\frac{1}{2}\sin 2x\)
(d) none of these
Solution
Use the complementary angle identities:
\[ \sin\left(\frac{\pi}{2}+x\right)=\cos x \]
\[ \cos\left(\frac{\pi}{2}-x\right)=\sin x \]
Therefore,
\[ \tan x \cdot \sin\left(\frac{\pi}{2}+x\right)\cdot \cos\left(\frac{\pi}{2}-x\right) = \tan x \cdot \cos x \cdot \sin x \]
Using
\[ \tan x=\frac{\sin x}{\cos x} \]
\[ = \frac{\sin x}{\cos x}\cdot \cos x \cdot \sin x \]
\[ = \sin^2 x \]
Since \(\sin^2 x\) is not equal to any of the options (a), (b), or (c),
the expression does not match the given choices.
Final Answer
\[ \boxed{\sin^2 x} \]
Hence, the correct option is (d) none of these.