# Tutor profile: Lucas F.

## Questions

### Subject: Pre-Calculus

Prove that $$ sec^2 \alpha\tan^2 \beta \ - \ tan^2\alpha \ sec^2\beta = tan^2\beta \ - \ tan^2 \alpha $$ .

Because $$ tan x = \frac{sinx}{cosx}$$ and $$sec x = \frac{1}{cosx}$$ From the left side of the equation, we have that: $$ sec^2 \alpha\tan^2 \beta \ - \ tan^2\alpha \ sec^2\beta = \frac{sec^2 \alpha\sin^2 \beta}{cos^2\beta} \ - \ \frac{tan^2\alpha}{cos^2\beta} $$ Factoring the numerator: $$ \frac{sec^2 \alpha\sin^2 \beta}{cos^2\beta} \ - \ \frac{tan^2\alpha}{cos^2\beta} = \frac{\frac{1}{cos^2\alpha} (sin^2\beta -sin^2\alpha)}{cos^2\beta}$$ We also have that: $$ sin^2x + cos^2 x = 1 $$ Therefore: $$ \frac{\frac{1}{cos^2\alpha} (sin^2\beta -sin^2\alpha)}{cos^2\beta} = \frac{sin^2\beta - 1 + cos^2\alpha}{cos^2\beta \ cos^2 \alpha} = \frac{-(1-sin^2\beta) + cos^2\alpha}{cos^2\beta \ cos^2 \alpha} = \frac{-(cos^2\beta) + cos^2\alpha}{cos^2\beta \ cos^2 \alpha}$$ Simplifying this expression: $$ \frac{-(cos^2\beta) + cos^2\alpha}{cos^2\beta \ cos^2 \alpha} = \frac{cos^2\alpha-cos^2\beta}{cos^2\beta \ cos^2 \alpha} = \frac{cos^2\alpha}{cos^2\beta \ cos^2 \alpha} - \frac{cos^2\beta}{cos^2\beta \ cos^2 \alpha} = \frac{1}{cos^2\beta} - \frac{1}{cos^2 \alpha}$$ Due to the identity: $$ \frac{-(cos^2\beta) + cos^2\alpha}{cos^2\beta \ cos^2 \alpha} = \frac{cos^2\alpha-cos^2\beta}{cos^2\beta \ cos^2 \alpha} = \frac{cos^2\alpha}{cos^2\beta \ cos^2 \alpha} - \frac{cos^2\beta}{cos^2\beta \ cos^2 \alpha} = \frac{1}{cos^2\beta} - \frac{1}{cos^2 \alpha}$$ $$ \frac{1}{cos^2\beta} - \frac{1}{cos^2 \alpha} = sec^2\beta - sec^2\alpha$$. Therefore: $$ sec^2 \alpha\tan^2 \beta \ - \ tan^2\alpha \ sec^2\beta = tan^2\beta \ - \ tan^2 \alpha $$.

### Subject: Biochemistry

Describe Western Blotting and its applications.

Western Blotting is a technique for identifying the presence of proteins in a mixture. It is semi-quantitative, meaning we can infer the relative amounts in each sample. The first step is separating the molecules according to size with SDS-PAGE. The samples are diluted in SDS, a detergent which gives them charge, and a reducing agent, which breaks their disulfide bonds. They are applied onto a polyacrylamide gel and an electric current makes them travel. Due to different speeds among larger and smaller molecules, they are separated. The proteins in the gel are then transferred onto a membrane, usually nitrocellulose or polyvinylidene fluoride (PVDF). These membranes are ideal for them to be exposed to antibodies which usually recognize them specifically. The last step is to visualize the signal, and this is usually done with secondary antibodies which are fluorescent or chemiluminescent and are able to attach to the primary antibodies (used in the previous step). We can thus visualize where the proteins are in the membrane and estimate their relative amount with the intensity of the signal. Western Blot is useful for a range of applications, from analyzing the expression of proteins in different conditions to examining the phosphorylation levels of enzymes.

### Subject: Biology

Why does the RNA contain Uracil while the DNA contains Thymine? Is there an evolutionary reason for that?

Uracil and Thymine are very similar molecules, with thymine having an extra methyl group. Both of them bind to Adenine. However, Uracil is the product of deamination of Cytosine, a process that occurs naturally. If the cell was not able to discriminate whether a U base is correct or a degraded C, there would be errors that compounded in the genetic material over time, causing harm to the organism. With the use of T instead of U, the cell has the ability to identify and correct degraded Cytosines.

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