Question 4 41 simplify without the use of a calculator...

QUESTION 4 4.1. Simplify without the use of a calculator (2cos(180\deg +x)sin(180\deg -x)sin74\deg )/(sin(x+360\deg )sin37\deg sin53\deg sin(x-90\deg )) 4.2. If 2sin\theta cos\theta =(2\sqrt(6))/(5),45\deg <\theta <90\deg Determine with the aid of diagram and without using a calculator the value of sin\theta 4.3. Prove the identity 4.3.1 4sin\theta cos^(3)\theta -4cos\theta sin^(3)\theta =sin4\theta 4.3.2 Hence, or otherwise solve the following . 1+4sin\theta cos^(3)\theta =4cos\theta sin^(3)\theta for \theta \epsi lon(0\deg ;360\deg )

Transcript

00:01 So for your questions, you actually have submitted too many in one problem, but i'd be happy to help you with this one proof.

00:08 And then you can resubmit the question and ask for just a particular portion of the questions.

00:15 Maybe you know how to do some and not the others.

00:18 And so let's do this proof.

00:22 So we're going to use the double angle identity for sign.

00:24 So that is 2 .sign x, cosine x plus sine x.

00:32 And in the denominator, we're going to use a double angle.

00:34 Angle identity for cosine.

00:36 We're going to use the two cosine squared of x minus one plus cosine x plus one equals that tangent.

00:47 Now we're going to factor in the numerator.

00:49 We're going to pull out that common factor of sine of x.

00:57 And in the denominator, we can see that these ones cancel, and we're going to factor out the cosine of x.

01:11 And now we can reduce this down and it becomes sine x over cosine of x.

01:18 And this becomes tangent of x by the tangent identity and let's see let's see we look at another question and see if i can do something quickly for you you have given sign of theta is equal to one -third you want to find i'll help you with that one so sign of theta is equal to one -third we know that tangent of theta is negative and we want to find cosine of theta and tangent of 2 theta.

02:06 Cosine of theta, and we want to find tangent of 2 theta.

02:13 So we know that sign is positive in the first and second quadrant, and tangent is negative in the second and third...

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