Question

tan (45° + a) +tan (45º – a)
cot (45° + a)+cot(45° – a)​

Answer 1

tan(45° + a) + tan(45° - a)

=> $\sf\dfrac{tan45 + tana}{1 - tan45.tana} + \dfrac{tan45 - tana}{1 + tan45tana}$

=> $\sf\dfrac{1 + tana}{1 - 1(tana)} + \dfrac{1 - tana}{1 + 1(tana)}$

=> $\sf\dfrac{1 + tana}{1 - tana} + \dfrac{1 - tana}{1 + tana}$

=> $\sf\dfrac{(1+tana)(1+tana) + (1-tana)(1-tana)}{(1+tana)(1-tana)}$

=> $\sf\dfrac{(1+tana)^{2} + (1-tana)^{2}}{1^{2} - tana^{2}}$

=> $\sf\dfrac{(1)^{2} + (tana)^{2} + 2tana + (1)^{2} + (tana)^{2} - 2tana}{1 - tan^{2}a}$

=> $\sf\dfrac{1 + tan^{2}a + 2tana + 1 + tan^{2}a - 2tana}{1 - tan^{2}a}$

=> $\sf\dfrac{2 + 2tan^{2}a}{1 - tan^{2}a}$

=> $\sf\dfrac{2(1 + tana)(1 + tana)}{(1 + tana)(1 - tana)}$

=> $\sf\dfrac{2 + 2tana}{1 - tana}$