- IN THIS VIDEO WE'LL USE DOUBLE 
ANGLE IDENTITIES TO SIMPLIFY
AND THEN EVALUATE 
THE FOLLOWING EXPRESSIONS.
SO THE FIRST EXPRESSION 
IS COSINE SQUARED PI
DIVIDED BY 12 - SINE SQUARED 
OF PI DIVIDED BY 12
WHICH FITS THE FORM 
OF THIS FIRST IDENTITY.
NOTICE ON THE RIGHT SIDE WE HAVE
COSINE SQUARED "A" - SINE 
SQUARED "A" = COSINE OF 2A.
SO FOR THE FIRST EXPRESSION, 
"A" IS EQUAL TO PI/12.
SO WE CAN REWRITE THIS AS COSINE 
OF 2A OR COSINE OF 2 x PI/12
PUT THE 2/1 THIS SIMPLIFIES.
SO WE HAVE A COSINE PI/6 
WHICH IS A 30 DEGREE ANGLE.
LETS GO AHEAD AND SKETCH 
OUR REFERENCE TRIANGLE
AND THE COSINE ON 30 DEGREES 
IS SQUARE ROOT 3 DIVIDED BY 2.
FOR THE SECOND EXPRESSION,
WE HAVE 2 COSINE OF PI/4 x 
COSINE PI/4 WHICH FITS
THIS DOUBLE ANGLE IDENTITY 
FOR SINE.
NOTICE WE HAVE 2 SINE "A" COSINE 
"A" ON THE RIGHT SIDE
WHICH IS EQUAL TO SINE OF 2A.
SO "A" IS EQUAL TO PI/4.
SO THIS IS EQUAL TO SINE 
OF 2A OR 2 x PI/4.
AGAIN, THIS SIMPLIFIES NICELY.
SO THIS EQUALS SINE 
OF PI/2 WHICH IS EQUAL TO 1
AND WE CAN VERIFY THIS IF WE 
NEED TO ON THE UNIT CIRCLE.
HERE'S PI/2 RADIANS 
ON THE UNIT CIRCLE.
SINE THETA IS EQUAL TO Y.
SO SINE PI/2 IS EQUAL TO 1.
IN OUR LAST EXAMPLE, WE HAVE 2 
COSINE SQUARED PI/2 - 1
WHICH FITS THE FORM 
OF THIS IDENTITY HERE
WHERE "A" = PI/2
AND SO THIS RIGHT SIDE IS EQUAL 
TO COSINE OF 2 x "A"
SO WE HAVE COSINE OF 2 x PI/2 
IS JUST GOING TO BE PI
AND COSINE OF PI IS EQUAL TO -1
AND THEN AGAIN, 
IF WE NEED TO WE CAN VERIFY
THIS ON THE UNIT CIRCLE.
HERE'S THE TERMINAL SIDE 
OF PI RADIANS.
COSINE THETA IS EQUAL TO X.
SO WE HAVE A COSINE FUNCTION 
VALUE OF -1.
