Verify: (1+cos squared theta)/(sin squared theta) = 2 csc squared theta - 1

Answer 1

Verify: (1+cos squared theta)/(sin squared theta) = 2 csc squared theta – 1

Verify: (1 + cos² θ)/sin² θ = 2 csc² θ - 1

Verify by starting at Left Hand Side ((1 + cos² θ)/sin² θ) and arrive to Right Hand Side (2 csc² θ - 1)

Known Trigonometry Facts: 1 + cot² θ = csc² θ and cot² θ = csc² θ - 1

Left Hand Side:

(1 + cos² θ)/sin² θ = 1/sin² θ + cos² θ/sin² θ

= csc² θ + cot² θ

= csc² θ + (csc² θ – 1)

= csc² θ + csc² θ – 1

= 2 csc² θ – 1 <== Right Hand Side

Answer 2

We can limit our manipulations using trig identities to the expression on the left hand side as follows:

[ 1 + cos²Θ ] / (sin²Θ) (split this into 2 separate fractions by dividing each term in numerator):

= 1 / (sin²Θ) + (cos²Θ) / (sin²Θ) (use a reciprocal id for 1st fraction, a quotient id for 2nd):

= csc²Θ + cot²Θ (use a pythagorean id for 2nd term):

= csc²Θ + (csc²Θ - 1) (gather like terms):

= 2csc²Θ - 1 ◊