Question

Verify that [tan(theta) + cot(theta)]^2 = sec^2(theta) + csc^2(theta)

Answer 1

(tanθ + cotθ)² = sec²θ + csc²θ

Expand left side: tan²θ + 2tanθcotθ + cot²θ

Evaluate middle term: 2tanθcotθ = $2*\frac{sin\theta}{cos\theta}*\frac{cos\theta}{sin\theta}$ = 2

⇒ tan²θ + 2+ cot²θ

= tan²θ + 1 + 1 + cot²θ

Apply trig identity:  tan²θ + 1 = sec²θ

⇒ sec²θ + 1 + cot²θ

Apply trig identity:  1 + cot²θ = csc²θ

⇒ sec²θ + csc²θ

Left side equals Right side so equation is verified