Question

Prove the following 2 trig identities. Show all steps!

Answer 1

a) multiply by cos²/cos², move sin/cos inside parentheses, simplify

d) multiply by (cot+cos); use cot=cos·csc, csc²-1=cot² in the denominator

Explanation:

You want to prove the identities ...

• sin²(x)(cot(x) +1)² = cos²(x)(tan(x) +1)²
• cos(x)cot(x)/(cot(x)-cos(x) = (cot(x)+cos(x)/(cos(x)cot(x))

Identities

Usually, we want to prove a trig identity by providing the steps that transforms one side of the identity to the expression on the other side. Here, each of these identity expressions can be simplified, so it is actually much easier to simplify both expressions to one that is common.

a) sin²(x)(cot(x) +1)² = cos²(x)(tan(x) +1)²

We are going to use s=sin(x), c=cos(x), (s/c) = tan(x), and (c/s) = cot(x) to reduce the amount of writing we have to do.

`s^2\left((c)/(s)+1\right)^2=c^2\left((s)/(c)+1\right)^2\qquad\text{given}\\\\\\(s^2(c+s)^2)/(s^2)=(c^2(s+c)^2)/(c^2)\qquad\text{use common denominator}\\\\\\(c+s)^2=(c+s)^2\qquad\text{cancel common factors; Q.E.D.}`

d) cos(x)cot(x)/(cot(x)-cos(x) = (cot(x)+cos(x)/(cos(x)cot(x))

Using the same substitutions as above, we have ...

`(c(c/s))/((c/s)-c)=((c/s)+c)/(c(c/s))\qquad\text{given}\\\\\\(c^2)/(c(1-s))=(c(1+s))/(c^2)\qquad\text{multiply num, den by s}\\\\\\(c(1+s))/((1-s)(1+s))=(c(1+s))/(c^2)\\\\\\(c(1+s))/(1-s^2)=(c(1+s))/(c^2)\\\\\\(c(1+s))/(c^2)=(c(1+s))/(c^2)\qquad\text{Q.E.D.}`

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Additional comment

The key transformation in (d) is multiplying numerator and denominator by (1+sin(x)). You can probably prove the identity just by doing that on the left side, then rearranging the result to make it look like the right side.

For (a), the key transformation seems to be multiplying by cos²(x)/cos²(x) and rearranging.

Sometimes it seems to take several tries before the simplest method of getting from here to there becomes apparent. The transformations described in the top "Answer" section may be simpler than those shown in the "Step-by-step" section.