# Thread: Proving identities

1. ## Proving identities

hello, i am having problems with my homework, theres five problems i can not figure out, any help would be great,

1) cot x * sec x * sin x = 1

2) 2sec^2 y =1/(1 - sin y) + 1/(1 + sin y)

3) sec^2 a + csc^2 a = sec^2 a * csc^2 a

4) sin^4 β - cos^4 β = 1 - 2cos^2 β

5) cos Φ/(1 + sinΦ) + (1 + sin Φ)/cos Φ = 2sec Φ

2. Originally Posted by jpenny32
hello, i am having problems with my homework, theres five problems i can not figure out, any help would be great,

1) cot x * sec x * sin x = 1

2) 2sec^2 y =1/(1 - sin y) + 1/(1 + sin y)

3) sec^2 a + csc^2 a = sec^2 a * csc^2 a

4) sin^4 β - cos^4 β = 1 - 2cos^2 β

5) cos Φ/(1 + sinΦ) + (1 + sin Φ)/cos Φ = 2sec Φ

1) $\displaystyle L.H.S. = cot x \times sec x \times sin x = \frac{cosx}{sinx} \times \frac{1}{cosx} \times sinx = 1$
2) $\displaystyle R.H.S = \frac{1}{1-siny} + \frac{1}{1+siny} = \frac{(1+siny)+(1-siny)}{(1-siny)(1+siny)}$
$\displaystyle = \frac{2}{(1-sin^{2}y)}= \frac{2}{cos^{2}y} = 2sec^{2}y$
3)$\displaystyle L.H.S. = sec^{2}a + csc^{2}a = \frac{1}{cos^{2}a} + \frac{1}{sin^{2}a} = \frac{sin^{2}a + cos^{2}a}{cos^{2}a \times sin^{2}a}$
= $\displaystyle \frac{1}{cos^{2}a \times sin^{2}a} = \frac{1}{cos^{2}a } \times \frac{1}{sin^{2}a} = sec^{2}a \times csc^{2}a$