Question

A true statement among the following identities is

• A. \(\sin5\theta=16\cos^4\theta\sin\theta-12\cos^2\theta\sin\theta+\sin\theta\)
• B. \(\sin5\theta=16\cos^4\theta-12\cos^2\theta+1\)
• C. \(\sin5\theta=16\cos^4\theta\sin\theta+12\cos^2\theta\sin\theta-\sin\theta\)
• D. \(\sin5\theta=16\cos^4\theta\sin\theta+12\cos^2\theta\sin\theta+\sin\theta\)

Hint:

To transform multiple-angle trigonometric formulas, use: \[ \sin^2\theta=1-\cos^2\theta \] This helps rewrite expressions entirely in terms of \(\cos\theta\) and \(\sin\theta\).

Answer 1

The Correct Option is A

Step 1: Recall the standard formula for \(\sin5\theta\).
We know that \[ \sin5\theta = 5\sin\theta -20\sin^3\theta +16\sin^5\theta \] We now express everything in terms of \(\cos\theta\).

Step 2: Use the identity \[ \sin^2\theta=1-\cos^2\theta \] First, \[ \sin^3\theta = \sin\theta(1-\cos^2\theta) \] and \[ \sin^5\theta = \sin\theta(1-\cos^2\theta)^2 \] Substituting into the formula, \[ \sin5\theta = 5\sin\theta -20\sin\theta(1-\cos^2\theta) +16\sin\theta(1-\cos^2\theta)^2 \]

Step 3: Expand the expression.
First expand: \[ (1-\cos^2\theta)^2 = 1-2\cos^2\theta+\cos^4\theta \] Thus, \[ \sin5\theta = 5\sin\theta -20\sin\theta +20\cos^2\theta\sin\theta \] \[ +16\sin\theta -32\cos^2\theta\sin\theta +16\cos^4\theta\sin\theta \] Now combine like terms.
Coefficient of \(\sin\theta\): \[ 5-20+16=1 \] Coefficient of \(\cos^2\theta\sin\theta\): \[ 20-32=-12 \] Hence, \[ \sin5\theta = 16\cos^4\theta\sin\theta -12\cos^2\theta\sin\theta +\sin\theta \]

Step 4: Match with the given options.
The obtained identity exactly matches option (1).

Step 5: Final conclusion.
Therefore, \[ \boxed{ \sin5\theta = 16\cos^4\theta\sin\theta -12\cos^2\theta\sin\theta +\sin\theta } \] Hence, the correct option is (1).