Our first line of defense when solving trigonometric functions is using a familiar identity/formula such as the Pythagorean Identities or the Double Angle Formulas.  When we are unable to use an identity or formula we are able to square both sides of the equation and with further manipulation we are usually able to produce one of these identities thus simplifying our problem and making it easier to solve.

Tangent is positive in two quadrants: quadrants 1 and 3 (where sine and cosine have the same sign as each other)

Cosine is negative in two quadrants: quadrants 2 and 3 (the left quadrants, where x is negative).

The only quadrant that shares both qualities is quadrant 3.

When drawn from the origin, a line 45 degrees above (counterclockwise from) the positive x-axis lies in quadrant I. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant I composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies along the positive x-axis. Since the adjacent side and hypotenuse are both represented by positive numbers, the fraction A/H is positive. Therefore, cos 45 is positive.

When drawn from the origin, a line 135 degrees counterclockwise from the positive x-axis lies in quadrant II. Sine is defined as the ratio between the opposite side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant II composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The opposite side of this triangle lies above the x-axis, and is therefore represented by a positive number. Since the opposite side and hypotenuse are both represented by positive numbers, the fraction O/H is positive. Therefore, sin 135 is positive.

When drawn from the origin, a line 135 degrees counterclockwise from the positive x-axis lies in quadrant II. Tangent is defined as the ratio between the opposite and adjacent sides of a triangle. A right triangle can be drawn in quadrant II composed of any point on that line, the origin and a point on the x-axis. The opposite side of this triangle lies above the x-axis, and is therefore represented by a positive number. The adjacent side of this triangle lies on the negative x-axis, and is therefore represented by a negative number. Since the opposite side and adjacent side have opposite signs, the fraction O/A is negative. Therefore, tan 135 is negative.

When drawn from the origin, a line 225 degrees counterclockwise from the positive x-axis lies in quadrant III. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant III composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies on the negative x-axis, and is therefore represented by a negative number. Since the opposite side and hypotenuse have opposite signs, the fraction A/H is negative. Therefore, cos 225 is negative.

When drawn from the origin, a line 225 degrees counterclockwise from the positive x-axis lies in quadrant III. Sine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant III composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The opposite side of this triangle lies below the x-axis, and is therefore represented by a negative number. Since the opposite side and hypotenuse have opposite signs, the fraction O/H is negative. Therefore, sin 225 is negative.

When drawn from the origin, a line 225 degrees counterclockwise from the positive x-axis lies in quadrant II. Tangent is defined as the ratio between the opposite and adjacent sides of a triangle. A right triangle can be drawn in quadrant III composed of any point on that line, the origin and a point on the x-axis. The opposite side of this triangle lies below the x-axis, and is therefore represented by a negative number. The adjacent side of this triangle lies on the negative x-axis, and is therefore represented by a negative number. Since the opposite side and adjacent side have the same sign, the fraction O/A is positive. Therefore, tan 225 is positive.

When drawn from the origin, a line 45 degrees above (counterclockwise from) the positive x-axis lies in quadrant I. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. Secant is defined as the reciprocal of cosine; therefore, the cosine and secant of an angle have the same sign. A right triangle can be drawn in quadrant I composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies along the positive x-axis. Since the adjacent side and hypotenuse are both represented by positive numbers, the fraction A/H is positive. Therefore, cos 45 is positive, as is its reciprocal, sec 45.

When drawn from the origin, a line 135 degrees counterclockwise from the positive x-axis lies in quadrant II. Sine is defined as the ratio between the opposite side of a triangle and the hypotenuse of the triangle. Cosecant is defined as the reciprocal of sine, so the sine and cosecant of an angle must have the same sign. A right triangle can be drawn in quadrant II composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The opposite side of this triangle lies above the x-axis, and is therefore represented by a positive number. Since the opposite side and hypotenuse are both represented by positive numbers, the fraction O/H is positive. Therefore, sin 135 is positive, as is its reciprocal, csc 135.