Question

cosa/1+sins+1+sina/cosa=2seca​

Answer 1

Hi !

$\frac{1+ sinA}{cosA} + \frac{cosA}{1+ sinA}$

Cross multiplication,

$\color{brown}\frac{(1+sinA) ^{2} + cos ^{2}A }{(1+sinA)cosA}(1+sinA)cosA$

$\color{red}\frac{1 + 2sinA + sin ^{2}A + cos ^{2}A }{(1+sinA)cosA}$

we know that ,

sin²A + cos²A = 1

$\frac{1 + 1 + 2 sinA}{(1+sinA)cosA}$

$\color{blue}\frac{2 + 2sinA}{(1+sinA)(cosA)}$

$\color{green}\frac{2(1+sinA)}{(1+sinA)(cosA)}$

1 + sinA gets cancelled ,

$\frac{2}{cos}$

2* 1/cosA

• We know that , 1/cosA = secA

hence,

2*secA = 2secA = RHS

proved !

Answer 2

Answer:

$\frac{cosa}{1 + sina} + \frac{1 + sina}{cosa} \\ \\ = \frac{ {cos}^{2}a + ( {1 + sina})^{2} }{(1 + sina)cosa} \\ \\ = \frac{ {cos}^{2}a + 1 + {sin}^{2} a + 2sina }{(1 + sina)cosa} \\ \\ = \frac{2 + 2sina}{(1 + sina)cosa} \\ \\ = \frac{2}{cosa} \\ \\ = 2seca$