Simplify a trigonometric identity with shifted angles

Expert intro: These problems usually look longer than they are. The trick is to rewrite each shifted trig function using angle identities and then reduce carefully, one factor at a time.

Detailed walkthrough

We simplify each factor using standard identities.

Left-hand side

$\sin\left(\frac{\pi}{2}+x\right)=\cos x$

$\cos(\pi-x)=-\cos x$

$\cot\left(\frac{3\pi}{2}+x\right)=\cot\left(\frac{\pi}{2}+x\right)= -\tan x$

So the left-hand side becomes

$\cos x\cdot(-\cos x)\cdot(-\tan x)=\cos^2x\tan x$

Since $\tan x=\dfrac{\sin x}{\cos x}$,

$\cos^2x\tan x=\cos^2x\cdot\frac{\sin x}{\cos x}=\sin x\cos x$

Right-hand side

$\sin\left(\frac{\pi}{2}-x\right)=\cos x$

$\sin\left(\frac{3\pi}{2}-x\right)=-\cos x$

$\cot\left(\frac{\pi}{2}+x\right)=-\tan x$

So the right-hand side becomes

$\cos x\cdot(-\cos x)\cdot(-\tan x)=\sin x\cos x$

Both sides simplify to the same expression:

$\boxed{\sin x\cos x}$

So the identity is true.

💡 Pitfall guide

The biggest error here is sign confusion with shifted angles. It helps to write a few anchor identities before starting: $\sin(\pi/2+x)=\cos x$, $\cos(\pi-x)=-\cos x$, and $\cot(\pi/2+x)=-\tan x$. Another easy slip is cancelling too early before rewriting everything consistently.

🔄 Real-world variant

If the cotangent terms were replaced by tangent terms, the signs could change and the identity might fail unless the whole expression were adjusted. These shifted-angle problems are very sensitive to whether the angle is $+x$ or $-x$, especially around $\pi/2$ and $3\pi/2$.

🔍 Related terms

trigonometric identity, shifted angles, cotangent