In worksheet on establishing identities and simplification using trigonometric ratios of complementary angles we will solve various types of practice questions on trigonometric ratios of complementary angles. Here you will get 17 different types of questions on establishing identities and simplification using trigonometric ratios of complementary angles.

**1.** Prove that sin (90° - θ) sec θ + cos (90° - θ) csc θ = 2

**2.** Prove that sin θ cos (90° - θ) + cos θ sin (90° - θ) = 1

**3.** Prove that cos A cos (90° - A) - sin A sin (90° - A) = 0

**4.** Prove that \(\frac{cos θ}{sin (90° - θ)}\) + \(\frac{sin
θ}{cos (90° - θ)}\) = 2

**5.** Prove that (1 + tan^{2} θ) cos θ cos (90° - θ) = cot (90° -
θ)

**6.** Prove that \(\frac{cot θ}{tan (90° - θ)}\) + \(\frac{cos
(90° - θ) tan θ sec (90° - θ)}{sin (90° - θ) cot (90° - θ) csc (90° - θ)}\) = 2

**7.** Prove that \(\frac{sin θ}{sin (90° - θ)}\) + \(\frac{cos
θ}{cos (90° - θ)}\) = sec θ csc θ

**8.** Simplify: \(\frac{sin 20°}{sin 70°}\) + \(\frac{cos
θ}{sin (90° - θ)}\)

**9.** Simplify: \(\frac{cos 70°}{sin 20°}\) + \(\frac{cos 59°}{sin
31°}\) - 8 cos^{2} 60°

**10.** Simplify: \(\frac{cos^{2} 20° + cos^{2} 70°}{sin^{2} 59°
+ sin^{2} 31°}\) + sin 35° sec 55°

**11.** Simplify: \(\frac{sin 80°}{cos 10°}\) + cos 59° csc 31°

**12.** Simplify: \(\frac{tan (90° - θ) cot θ}{sec (90° - θ) csc
θ}\)

**13.** Simplify: sin^{2} A – cos^{2} B + sin^{2} (90° - A) – cos^{2}
(90° - B)

**14.** If A + B = 90°, sin A= a and sin B = b then prove that
a^{2} + b^{2} = 1.

**15.** In ∆ABC, prove that cos \(\frac{B + C}{2}\) = sin \(\frac{A}{2}\)

**16.** In ∆ABC, prove that tan \(\frac{A + B}{2}\) = cot \(\frac{C}{2}\)

**17.** In ∆ABC, prove that sec \(\frac{C + A}{2}\) = csc \(\frac{B}{2}\)

Answers on Worksheet on Establishing Identities and Simplification Using Trigonometric Ratios of Complementary Angles are given below to check the exact answers of the questions.

**Answers:**

**8.** 2

**9.** 0

**10.** 2

**11.** 2

**12.** cos^{2} θ

**13.** 0.

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