Simplify \$(("sin"^2\theta) /("cos"\theta)) + (("cos"^2\theta) /("sin"\theta))\$.

The given values expression is

\$(("sin"^2\theta) /("cos"\theta)) + (("cos"^2\theta) /("sin"\theta))\$

To simplify the given expression, let’s first express \$sin^2θ\$ and \$cos^2θ\$ in terms of sin θ and cos θ:

\$ sin^2 θ = 1 - cos^2 θ \$ (using the Pythagorean identity)

\$ cos2 θ = 1 - sin^2 θ \$ (using the Pythagorean identity)

Now, substitute these values into the expression

\$(("sin"^2\theta) /("cos"\theta)) + (("cos"^2\theta) /("sin"\theta))\$

⇒ \$(1 - cos^2θ) /(cosθ) + (1 - sin^2θ) /(sinθ)\$

⇒ \$1/(cosθ) - (cos^2θ)/(cosθ) + 1/(sinθ) - (sin^2θ)/(sinθ)\$

⇒ \$1/(cosθ) - cosθ + 1/(sinθ) - sinθ\$

⇒ \$secθ - cosθ + cscθ - sinθ\$

So, the simplified expression is \$secθ − cosθ + cscθ − sinθ\$.

Option A simplifies the expression by using trigonometric identities, substituting and combining terms for simplification

\$secθ - cosθ + cscθ - sinθ\$

This choice changes the order of terms, making it different from the original expression

\$cosθ + secθ + sinθ + cscθ\$

This choice switches the signs of all terms, which is incorrect

\$cosθ − secθ + sinθ − cscθ \$

This choice rearranges the terms, which is different from the original expression

\$secθ + cosθ − cscθ + sinθ\$

Option

A

Can you briefly tell me what you’ve learned and understood in today’s lesson?