Prove and apply trigonometric identities involving sin, cos, tan, and their reciprocals, building on the fundamental identity.
sin²θ+cos²θ=1, 1+tan²θ=sec²θ, and 1+cot²θ=cosec²θ are the three core identities used to simplify and prove trigonometric expressions.
Simplify: sec²θ − tan²θ
By the identity 1+tan²θ=sec²θ, this equals 1
Prove: (1−cos²θ)=sin²θ
Directly from sin²θ+cos²θ=1, rearranging gives 1−cos²θ=sin²θ
Proving identities involves manipulating one side (usually the more complex one) using known identities until it matches the other side.
Prove: sinθ/cosθ = tanθ (by definition, but express using identities)
This is the definition of tanθ, confirming sinθ/cosθ=tanθ directly
Simplify: (1+tanθ)² + (1−tanθ)²
=1+2tanθ+tan²θ+1−2tanθ+tan²θ=2+2tan²θ=2sec²θ
1. Simplify: cosec²θ − cot²θ
2. Prove: 1−sin²θ=cos²θ
3. Simplify: (secθ−tanθ)(secθ+tanθ)
4. Simplify: sin²θ×cosec²θ
5. Simplify: tan²θ − sec²θ