ML Aggarwal Solutions Class 10 Mathematics Solutions for Trigonometric Identities Exercise 18 in Chapter 18 - Trigonometric Identities

Question 35 Trigonometric Identities Exercise 18

If sin θ + cos θ = √2 sin (90° – θ), show that cot θ = √2 + 1

Solution:

In RHS, first we can apply complementary trignometric angles concept

Given, sin θ + cos θ = √2 sin (90° – θ)

sin θ + cos θ = √2 cos θ

On dividing by sin θ, we have

1 + cot θ = √2 cot θ

1 = √2 cot θ – cot θ

(√2 – 1) cot θ = 1

cot θ = 1/ (√2 – 1)

=\frac{1 \times(\sqrt{2}+1)}{(\sqrt{2}-1)(\sqrt{2}+1)} (Rationalising the denominator)

\begin{aligned} &=\frac{(\sqrt{2}+1)}{(\sqrt{2})^{2}-(1)^{2}}=\frac{\sqrt{2}+1}{2-1}=\frac{\sqrt{2}+1}{1} \\ &=\sqrt{2}+1=\text { R.H.S. } \end{aligned}

Hence, cot θ = √2 + 1

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