Question

Prove that:
tanθ +cot θ = 1/sinθ cos θ

Answer 1

To prove that tanθ + cotθ = 1/sinθ cosθ, we use the definitions of tangent and cotangent, find a common denominator, and apply the Pythagorean identity to arrive at the desired expression.

Step-by-step explanation:

To prove that tanθ + cotθ = 1/sinθ cosθ, let's use the definitions of the trigonometric functions and manipulate the terms.

• The tangent of an angle (θ) is equal to the ratio of the sine to the cosine of that angle: tanθ = sinθ / cosθ.
• The cotangent of an angle (θ) is the reciprocal of the tangent, which can also be expressed as the ratio of the cosine to the sine of that angle: cotθ = cosθ / sinθ.

Now, let's add them up:

tanθ + cotθ = (sinθ / cosθ) + (cosθ / sinθ)

We will find a common denominator to combine these fractions:

= (sin^2θ + cos^2θ) / (sinθ cosθ)

Using the Pythagorean identity, which states that sin^2θ + cos^2θ = 1, we can simplify the numerator:

= 1 / (sinθ cosθ)

Thus, we've shown that tanθ + cotθ is indeed equal to 1/sinθ cosθ.