5 sin^2(θ) − 6csc^2(θ) + 5cos^2(θ) + 6cot^2(θ) is equivalent to...

Answer 1

Have your trig identities on hand for this, they make spotting out little tricks easier to do.

Usually it's a good idea to get most of the problem in terms of sine and cosine functions. A good start is trying to eliminate the cotangent function.

Let's find a term of our expression we can use to do that. We do have a co-secant function.

Recall: csc²(θ) = 1 + cot²(θ)

Let's substitute that alternative expression in place of our co-secant:

5sin²(θ) - 6 * (1 + cot²(θ) ) + 5cos²(θ) + 6cot²(θ)

Now distribute it out:

5sin²(θ) - 6 - 6cot²(θ) + 5cos²(θ) + 6cot²(θ)

We see the 6cot²(θ) cancels out. Now we have:

5sin²(θ) - 6 + 5cos²(θ)

Now let's see if we can put the expression in terms of either sine or cosine functions. For this problem, I will get everything in terms of cosine, so we will try to put our sine function in terms of an expression made up of cosine functions.

Recall:

sin²(θ) = 1 - cos²(θ)

Let's substitute that in place of our sine function now:

5*(1-cos²(θ) ) - 6 + 5cos²(θ)

Distribute it out:

5 - 5cos²(θ) -6 + 5cos²(θ)

Now we can cancel out our cosine terms, leaving us with simple arithmetic:

5 - 6

That means our answer is -1. Not too scary, just a little bit of algebraic manipulation. That's why it's always nifty to memorize these important trig identities to see past the puzzles.

Hope this helped!

Answer 2

Rearrange:

5 sin^2 £ + 5 cos^2 £ + 6 cot^2 £ - 6 csc^2 £

factor

5 ( sin^2 £ + cos^2 £ ) - 6 ( -cot^2 £ + csc^2 £)

identities

5( 1) -6( 1)

5-6 = -1