- IN THESE NEXT TWO EXAMPLES
WE'RE ASKED TO REWRITE 
A TRIG EXPRESSION
USING A HALF ANGLE IDENTITY.
HERE WE HAVE SINE 
OF 5X RAISED TO THE 2nd POWER
SO WE'LL BE APPLING THE HALF 
ANGLE IDENTITY FOR SINE.
SO THERE'S A COUPLE OF THINGS 
WE HAVE TO REMEMBER HERE.
FIRST, NOTICE HOW THIS FUNCTION 
IS BEING SQUARED
SO IT'S NOT A PERFECT FIT.
IF WE WANT TO REWRITE A SQUARE 
ROOT USING A RATIONAL EXPONENT,
REMEMBER THE SQUARE ROOT OF X IS 
EQUAL TO X TO THE POWER OF 1/2.
SO LETS START BY TAKING 
THIS HALF ANGLE IDENTITY
AND SQUARING BOTH SIDES
AND WE'LL ALSO REWRITE 
THIS SQUARE ROOT
USING A RATIONAL EXPONENT
SO WE'LL HAVE + OR - THIS WILL 
BE 1 - COSINE "A" DIVIDED BY 2
RAISED TO THE 1/2 POWER SQUARED.
THE NEXT RULE WE HAVE 
TO REMEMBER
IS WHEN WE HAVE 
A POWER TO A POWER
WE MULTIPLY OUR EXPONENTS.
SO 1/2 x 2 IS GOING TO EQUAL 1 
AND SINCE WE'RE SQUARING THIS,
THIS + OR - GOES AWAY
BECAUSE IF WE SQUARE A POSITIVE 
IT'S POSITIVE
AND IF WE SQUARE A NEGATIVE 
IT'S STILL POSITIVE.
SO WE HAVE THE SINE 
OF A/2 SQUARED
IT'S GOING TO BE EQUAL TO 1 
- COSINE "A" DIVIDED BY 2.
SO THE NEXT STEP IS TO DETERMINE
WHAT IS THE VALUE OF "A" 
GIVEN SINE OF 5X SQUARED.
WELL 5X HAS TO EQUAL A/2.
SO IF WE SET A/2 = 5X
AND MULTIPLY BOTH SIDES BY 2 WE 
KNOW THAT "A" IS EQUAL TO 10X.
SO USING THE HALF 
ANGLE IDENTITIES
"A" WILL ALWAYS BE 
TWICE THE GIVEN ANGLE.
SO NOW WE HAVE ALL 
THE INFORMATION
WE NEED TO APPLY 
THIS SLIGHT VARIATION
OF THE HALF ANGLE IDENTITY 
FOR SINE.
WE'RE GOING TO HAVE SINE 
OF 5X RAISED TO THE 2nd
IT'S GOING TO BE EQUAL 
TO 1 - COSINE 10X DIVIDED BY 2
AND IF WE WANT WE CAN REWRITE 
THIS AS 1/2 - 1/2 COSINE 10X.
SO THERE'S REALLY 
NOT A LOT GOING ON HERE
OTHER THAN GETTING FAMILIAR
WITH THESE NEW 
HALF ANGLE IDENTITIES.
LET'S TAKE A LOOK AT ONE MORE.
HERE WE HAVE COSINE OF 2X RAISED 
TO THE 4th POWER
SO LET'S START BY TAKING THIS 
HALF ANGLE IDENTITY FOR COSINE
AND RAISING IT TO THE 4th POWER.
SO WE'LL HAVE COSINE OF A/2
AND WE'RE GOING TO RAISE IT 
TO THE 4th POWER THIS TIME.
IT'S GOING TO BE EQUAL TO + OR -
AND WE'LL ALSO WRITE THIS 
AS A RATIONAL EXPONENT AGAIN
SO WE'LL HAVE + OR - THIS WILL 
BE 1 + COSINE "A" DIVIDED BY 2
TO THE 1/2 POWER AND THEN AGAIN 
WE'RE RAISING BOTH SIDES
TO THE 4th POWER.
WELL HERE WE'RE GOING TO HAVE 
1/2 x 4 THAT'S GOING TO BE 2.
SO WE HAVE COSINE 
OF A/2 RAISED TO THE 4th
IT'S GOING TO BE EQUAL TO 1 + 
COSINE "A" ALL OVER 2
RAISED TO THE 2nd POWER.
AGAIN, NOTICE HOW THE + OR - 
SYMBOL IS NO LONGER NEEDED
BECAUSE WE'RE RAISING IT 
TO AN EVEN POWER
AND REMEMBER 
TO SQUARE A FRACTION
WE CAN SQUARE THE DENOMINATOR 
AND SQUARE THE NUMERATOR.
SO LET'S GO AHEAD 
AND MULTIPLY THIS OUT.
OUR DENOMINATOR IS GOING 
TO BE 4.
IF WE MULTIPLY THIS OUT,
WE'RE GOING TO HAVE 1 
+ COSINE "A" x 1 + COSINE "A"
SO WE'LL HAVE 1 + COSINE "A" 
+ COSINE "A."
THAT'S 2 COSINE "A"
AND THEN WE'LL HAVE COSINE 
SQUARED "A."
NOW WE'LL GO AHEAD
AND APPLY THIS SLIGHT VARIATION 
OF THE HALF ANGLE IDENTITY
TO THE GIVEN EXPRESSION
AND REMEMBER IF A/2 
IS EQUAL TO 2X
THE ANGLE 
AND THE HALF ANGLE IDENTITY
IF WE MULTIPLY BOTH SIDES BY 2 
WE'LL HAVE "A" = 4X.
SO WHEN WE APPLY THIS IDENTITY
WE'LL HAVE COSINE OF 2X RAISED 
TO THE 4th POWER.
IT'S GOING TO BE EQUAL TO 1 + 2 
COSINE 4X + COSINE SQUARED 4X
ALL DIVIDED BY 4
AND WE COULD FACTOR OUT THE 1/4 
TO HAVE 1/4 x THE QUANTITY
1 + 2 COSINE 4X 
+ COSINE SQUARED 4X
AND OF COURSE, WE COULD 
ALSO DISTRIBUTE THE 1/4
BUT I THINK WE'LL GO AHEAD 
AND LEAVE IT LIKE THIS.
AGAIN, OTHER THAN DOING SOME 
MANIPULATIONS
TO THESE HALF ANGLE IDENTITIES
THERE'S NOT A LOT GOING ON HERE
OTHER THAN BECOMING MORE 
FAMILIAR WITH THESE IDENTITIES.
