Could you help me solve this two trig equations?

Question

sinX=tanX

.5secX-1=0

Answer 1

Get everything on the left:

sin(x) - tan(x) = 0

Write tangent in terms of sin and cos:

sin(x) - sin(x)/cos(x) = 0

Factor out sin:

sin(x)[1 - 1/cos(x)] = 0

So either

sin(x) = 0, which happens at 0 degrees, 180 degrees (π radians), 360 degrees (2π radians), etc. This is written as kπ (all integer multiples of π).

or

1 - 1/cos(x) = 0

Add 1/cos(x) to both sides:

1 = 1/cos(x)

Multiply both sides by cos(x):

cos(x) = 1

cos(x) = 1 at 0, 360 degrees (2π), 520 degrees (4π), etc. This is written as 2kπ (all even integer multiples of π).

Since the two solutions "overlap", the second one becomes irrelevant, and the answer is "all integer multiples of 180 degrees or π radians" (or kπ).

.5sec(x) - 1 = 0

Add 1 to both sides:

.5sec(x) = 1

Divide both sides by .5:

sec(x) = 1/.5 = 2

Write sec(x) in terms of cos(x):

1/cos(x) = 2

Flip both sides:

cos(x) = 1/2

Since cos(x) = 1/2 at 60 degrees (π/3) and -60 degrees (-π/3), the answer would be π/3 + 2kπ or -π/3 + 2kπ.