What is the general solution for 1/((sec^2(a)))+1/((cos^2(a)))=1 ?

The general solution for 1/((sec^2(a)))+1/((cos^2(a)))=1 is No Solution for a\in\mathbb{R}

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1/((sec^2(a)))+1/((cos^2(a)))=1

\frac{1}{( sec ^{2} ( a ) )} + \frac{1}{( cos ^{2} ( a ) )} = 1

No Solution for a \in R

Solution steps

\frac{1}{( sec ^{2} ( a ) )} + \frac{1}{( cos ^{2} ( a ) )} = 1

Subtract

1

from both sides

\frac{1}{sec ^{2} ( a )} + \frac{1}{cos ^{2} ( a )} - 1 = 0

Simplify

\frac{1}{sec ^{2} ( a )} + \frac{1}{cos ^{2} ( a )} - 1 : \frac{cos ^{2} ( a ) + sec ^{2} ( a ) - sec ^{2} ( a ) cos ^{2} ( a )}{sec ^{2} ( a ) cos ^{2} ( a )}

\frac{cos ^{2} ( a ) + sec ^{2} ( a ) - sec ^{2} ( a ) cos ^{2} ( a )}{sec ^{2} ( a ) cos ^{2} ( a )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos^{2} (a) + sec^{2} (a) - sec^{2} (a) cos^{2} (a) = 0

Rewrite using trig identities

- 1 + \frac{1}{sec ^{2} ( a )} + sec^{2} (a) = 0

Solve by substitution

sec (a) = \frac{3}{2} + \frac{1}{2} i, sec (a) = - \frac{3}{2} - \frac{1}{2} i, sec (a) = - \frac{3}{2} + \frac{1}{2} i, sec (a) = \frac{3}{2} - \frac{1}{2} i

sec (a) = \frac{3}{2} + \frac{1}{2} i :

No Solution

sec (a) = - \frac{3}{2} - \frac{1}{2} i :

No Solution

sec (a) = - \frac{3}{2} + \frac{1}{2} i :

No Solution

sec (a) = \frac{3}{2} - \frac{1}{2} i :

No Solution

Combine all the solutions

No Solution for a \in R

(1 - cos (a)) (1 + cos (a)) = tan (a) sin (a)

tan^{2} (x) + \frac{1}{6} + \frac{tan ( 1 )}{3} = 0

- 2 cos^{2} (x) - 5 sin (x) + 5 = 0

3 - 4 sin^{3} (x) = sin^{3} (x)

cos^{4} (x) = \frac{3}{8} + \frac{1}{2} cos^{2} (x) + \frac{1}{8} cos^{4} (x)

What is the general solution for 1/((sec^2(a)))+1/((cos^2(a)))=1 ?
The general solution for 1/((sec^2(a)))+1/((cos^2(a)))=1 is No Solution for a\in\mathbb{R}