Question

the expression ( 12 sin ^ { 2 } x - 12 ) is equivalent to
answer
( - 12 sin ^ { 2 } x )( - 12 cos ^ { 2 } x )( - 12 \tan ^ { 2 } x )( - 12 csc ^ { 2 } x )( - 12 sec ^ { 2 } x )( - 12 cot ^ { 2 } x )( 12 sin ^ { 2 } x )( 12 cos ^ { 2 } x )( 12 \tan ^ { 2 } x )( 12 csc ^ { 2 } x )( 12 sec ^ { 2 } x )( 12 cot ^ { 2 } x )( - 12 )( 12 )

Explanation:

Step1: Factor out the common factor

Factor out \(12\) from the expression \(12\sin^{2}x - 12\).
\(12(\sin^{2}x - 1)\)

Step2: Use the Pythagorean identity

Recall the Pythagorean identity \(\sin^{2}x+\cos^{2}x = 1\), which can be rewritten as \(\sin^{2}x-1=-\cos^{2}x\).
Substitute \(\sin^{2}x - 1=-\cos^{2}x\) into \(12(\sin^{2}x - 1)\).
\(12\times(-\cos^{2}x)=- 12\cos^{2}x\)

Answer:

\(-12\cos^{2}x\)