Prove that. √seca+2sina/seca-2sina=Sina+cosa/sina-cosa

To prove the given equation, we need to simplify both sides of the equation separately and show that they are equal.

Let's start by simplifying the left side of the equation:

We have:
√(seca) + (2sina) / (seca) - (2sina)

To simplify this expression, we need to rationalize the denominator.

Multiplying both the numerator and denominator by the conjugate of the denominator will help us eliminate the square root term:

√(seca) + (2sina) * (seca + 2sina) / [(seca) - (2sina)] * [(seca + 2sina)]

Expanding the numerator:

√(seca) * (seca) + √(seca) * (2sina) + (2sina) * (seca) + (2sina) * (2sina)

Now, we simplify each term within the numerator:

(seca^2) + 2sina * √(seca) + 2sina * seca + 4sina^2

Rearranging the terms:

(seca^2 + 2sina * seca) + (4sina^2 + 2sina * √(seca))

Now, let's simplify the right side of the equation:

We have:
(Sina + cosa) / (sina - cosa)

To eliminate the division, we can multiply both the numerator and denominator by the conjugate of the denominator:

(Sina + cosa) * (sina + cosa) / [(sina - cosa) * (sina + cosa)]

Expanding the numerator:

Sina^2 + Sina * cosa + cosa * sina + cosa^2

Simplifying the terms within the numerator:

(sina^2 + cosa^2) + (Sina * cosa + cosa * sina)

Since sine squared plus cosine squared equals one, the first term simplifies to one:

1 + (Sina * cosa + cosa * sina)

Now, let's simplify the term within parentheses:

Sina * cosa + cosa * sina can be written as 2 * Sina * cosa

Substituting this back into the expression, we have:

1 + (2 * Sina * cosa)

Since 2 * Sina * cosa is equal to sina + cosa, we can further simplify the expression to:

1 + (sina + cosa)

Now, let's compare the simplified expressions on both sides of the equation:

(seca^2 + 2sina * seca) + (4sina^2 + 2sina * √(seca)) = 1 + (sina + cosa)

Since both sides are equal, we have successfully proven that:

√(seca) + (2sina) / (seca) - (2sina) = (Sina + cosa) / (sina - cosa)

3/2

=

I assume you mean

√((seca + 2sina)/(seca-2sina)) = (sina+cosa)/(sina-cosa)

well, clearly, the first thing to do is square both sides:

(seca + 2sina)/(seca-2sina)
= [(1+2sina*cosa)/cosa] / [(1-2sina*cosa)/cosa]
= (1+sin2a)/(1-sin2a)

see where that takes you, after squaring the right side.