Simplifying a cotangent sine expression using secant

Key concept: The expression j(x)=((cot x)(sin x))/sin^2(x-1) is best handled by reducing the cotangent-sine product before worrying about the denominator [1].

Step by step

Key idea

The factor $\cot x\cdot \sin x$ is the first place to simplify in $j(x)=\dfrac{(\cot x)(\sin x)}{\sin^2(x-1)}$. Using the identity $\cot x=\dfrac{\cos x}{\sin x}$, we get

$(\cot x)(\sin x)=\left(\frac{\cos x}{\sin x}\right)(\sin x)=\cos x.$

So the expression becomes

$j(x)=\frac{\cos x}{\sin^2(x-1)}.$

If the task specifically asks for an expression involving $\sec x$ and no other trigonometric functions, then the numerator can be rewritten as $\cos x=\dfrac{1}{\sec x}$. That gives

$j(x)=\frac{1/\sec x}{\sin^2(x-1)}=\frac{1}{\sec x\,\sin^2(x-1)}.$

Method steps

Start with the definition of cotangent:

$\cot x=\frac{\cos x}{\sin x}.$

Then multiply by $\sin x$ to cancel the denominator. This is the cleanest algebraic simplification in the whole expression, because it removes one trigonometric ratio immediately.

After that, decide how to express the remaining cosine. Since the question requests $\sec x$, use the reciprocal identity

$\sec x=\frac{1}{\cos x}, \quad \text{so} \quad \cos x=\frac{1}{\sec x}.$

That leaves an answer with secant only in the simplified numerator.

Final form and identity check

The simplified expression is

$\boxed{j(x)=\frac{1}{\sec x\,\sin^2(x-1)}}.$

A useful check is to verify that no extra trigonometric functions were introduced unnecessarily. The original product $(\cot x)(\sin x)$ collapses to $\cos x$, and $\cos x$ is exactly what can be rewritten with $\sec x$. The denominator $\sin^2(x-1)$ remains unchanged because the prompt only asks to rewrite the expression, not to expand or convert the shift $(x-1)$.

Common mistake to avoid

A frequent error is to replace $\cot x$ with $\sec x$. Those are not equivalent: $\cot x$ is a quotient of cosine and sine, while $\sec x$ is the reciprocal of cosine. Another mistake is to stop after getting $\cos x/\sin^2(x-1)$ even when the instruction asks for secant. The correct final step is to rewrite $\cos x$ as $1/\sec x$, not to force secant into the denominator of the shifted sine term.

If your teacher expects a fully trig-simplified form, keep the denominator as written and only transform the numerator. That respects the original structure while satisfying the requested function form.

Pitfall alert

The place where this expression usually goes wrong is the product $(\cot x)(\sin x)$, because many students try to simplify the denominator first and never notice that the numerator collapses instantly. Since $\cot x=\cos x/\sin x$, the $\sin x$ cancels cleanly, but if you forget that cancellation you may carry extra factors all the way through and end up with a much messier answer. Another real trap is mixing up reciprocal identities. $\sec x$ is the reciprocal of cosine, not sine, so writing $\sin x=1/\sec x$ would be incorrect. Also, do not change $\sin^2(x-1)$ into something like $\sin^2 x-1$; the square applies to the entire shifted angle. Keep the shift intact unless the question explicitly asks for expansion. If the final instruction says “involving sec x,” one safe route is to simplify to $\cos x/\sin^2(x-1)$ first and then rewrite $\cos x$ as $1/\sec x$.

Try different conditions

If the numerator were changed to $(\tan x)(\cos x)$ instead of $(\cot x)(\sin x)$, the rewritten problem would be: $j(x)=\dfrac{(\tan x)(\cos x)}{\sin^2(x-1)}$, and the same strategy would apply, but with a different identity. Since $\tan x=\sin x/\cos x$, the product $(\tan x)(\cos x)$ simplifies to $\sin x$, not $\cos x$. Then the expression would become $\dfrac{\sin x}{\sin^2(x-1)}$. If the target still asked for secant only, you would need to decide whether the expression can even be rewritten naturally in terms of secant alone; in many cases it cannot without introducing additional identities or converting back through cosine first. Another useful variation is to replace the denominator with $\cos^2(x-1)$. Then the expression would be more naturally rewritten using $\sec(x-1)$, showing how the requested target function depends on which trig factor survives after simplification.

Further reading

cotangent identity simplification, secant reciprocal identity, trigonometric expression rewriting