Problems on Trigonometric Identities

We use the definition of tangent and cotangent to develop the left side of the equation

We use that and the definitions of secant and cosecant to obtain

which is what we wanted to reach.

First we develop the square

We factor from both addends, use the identity and the definition of cosecant and cotangent

We develop the right side, starting by factoring from both addends

We use the identity and the definition of secant

We use the definition of cotangent and secant

We cancel the factor and use the definition of cosecant

We develop with the definitions of secant and cosecant and operate the sum of fractions

Finally we use the identity and obtain the desired result

First we note that

The sine sum formula is

And using it we obtain the desired identity immediately

The definition of cotangent tells us that

We use the tangent sum formula and simplify

We divide the numerator and denominator by , then use cotangent to reduce the expression

We use the double angle sine formula

We consider that since then

We simplify and apply the definition of tangent

We substitute with and the double angle sine formula and perform the fraction multiplication operation

We develop and simplify

We use the formulas to convert from sums to products of trigonometric functions

Then

We simplify and use the definition of tangent. Also tangent is an odd function so