Question

examples: simplify and evaluate a trig expression
example: use double angle identities to
simplify and then evaluate.
\\( \cos ^ { 2 } \left( \frac { \pi } { 12 } \
ight) - \sin ^ { 2 } \left( \frac { \pi } { 12 } \
ight) = \cos \left( \frac { 2 \pi } { 12 } \
ight) = \cos \left( \frac { \pi } { 6 } \
ight) = \frac { \sqrt { 3 } } { 2 } \\)
\\( 2 \sin \left( \frac { \pi } { 4 } \
ight) \cos \left( \frac { \pi } { 4 } \
ight) = \sin \left( \frac { 2 \pi } { 4 } \
ight) = \sin \left( \frac { \pi } { 2 } \
ight) = 1 \\)
\\( 2 \cos ^ { 2 } \left( \frac { \pi } { 2 } \
ight) - 1 \\)
\\( \cos ( 2 a ) = \cos ^ { 2 } a - \sin ^ { 2 } a \\)
\\( \cos ( 2 a ) = 1 - 2 \sin ^ { 2 } a \\)
\\( \cos ( 2 a ) = 2 \cos ^ { 2 } a - 1 \\)
\\( \sin ( 2 a ) = 2 \sin a \cos a \\)
\\( \tan ( 2 a ) = \frac { 2 \tan a } { 1 - \tan ^ { 2 } a } \\)
simplify without a calculator:
\\( \cos ^ { 2 } \left( 15 ^ { \circ } \
ight) - \sin ^ { 2 } \left( 15 ^ { \circ } \
ight) \\)
\\( = \\)
\\( = \\)

Explanation:

Step1: Use double - angle identity

According to the double - angle identity for cosine, \(\cos(2A)=\cos^{2}A-\sin^{2}A\). Here \(A = 15^{\circ}\), so \(\cos^{2}(15^{\circ})-\sin^{2}(15^{\circ})=\cos(2\times15^{\circ})\).

Step2: Simplify the angle

Calculate \(2\times15^{\circ}=30^{\circ}\), so \(\cos(2\times15^{\circ})=\cos(30^{\circ})\).

Step3: Evaluate \(\cos(30^{\circ})\)

We know that \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\).

Answer:

\(\cos^{2}(15^{\circ})-\sin^{2}(15^{\circ})=\cos(30^{\circ})=\frac{\sqrt{3}}{2}\)