Solve tanx+cosx/1+sinx=secx
In two different ways
multiply top and bottom on the left by (1-sinx)
multiply both sides by 1+sinx.
To solve the equation tan(x) + cos(x)/(1 + sin(x)) = sec(x), we can approach it in two different ways. Here is the first method:
Method 1: Combining terms with a common denominator:
Step 1: Multiply both sides of the equation by (1 + sin(x)) to clear the denominator:
(1 + sin(x))(tan(x)) + cos(x) = sec(x)(1 + sin(x))
tan(x) + sin(x)tan(x) + cos(x) = sec(x) + sin(x)sec(x)
Step 2: Combine like terms:
sin(x)tan(x) + tan(x) + cos(x) = sec(x) + sin(x)sec(x)
Step 3: Rearrange the terms:
tan(x) + sin(x)tan(x) + cos(x) - sec(x) - sin(x)sec(x) = 0
Step 4: Factor out a common term:
tan(x)(1 + sin(x)) + cos(x)(1 - sin(x)) - sec(x)(1 + sin(x)) = 0
Step 5: Use the identity sec(x) = 1/cos(x) to simplify the equation further:
tan(x)(1 + sin(x)) + cos(x)(1 - sin(x)) - (1/cos(x))(1 + sin(x)) = 0
Step 6: Combine like terms and multiply through by cos(x):
tan(x) + sin(x)tan(x) + cos(x) - sin(x)cos(x) - 1 - sin(x) = 0
Step 7: Rearrange the terms:
tan(x) + sin(x)tan(x) - sin(x)cos(x) + cos(x) - sin(x) - 1 = 0
Step 8: Combine like terms:
tan(x)(1 + sin(x)) - sin(x)(cos(x) - 1) + cos(x) - 1 = 0
We have reached a simplified form of the equation. From here, you can use algebraic methods to solve for x.